# Chemical Power for Swarms of Microscopic Robots in Blood Vessels

Tad Hogg  Tad Hogg
Institute for Molecular Manufacturing
Palo Alto, CA

###### Abstract

Microscopic robots in the bloodstream could obtain power from fuel cells using
glucose and oxygen. Previous studies of small numbers of such robots operating
near each other showed how robots compete with their neighbors for oxygen.
However, proposed applications involve billions of such robots operating
throughout the body. With such large numbers, the robots can have systemic
effects on oxygen concentration. This paper evaluates these effects and their
consequences for robot power generation, oxygen available to tissue and
heating as such robots move with the blood. When robots consume oxygen as fast
as it diffuses to their surfaces, available power decreases significantly as
robots move from the lungs, through arteries to capillaries and veins. Tens of
billions of robots can obtain hundreds of picowatts throughout the circuit,
while a trillion robots significantly deplete oxygen in the veins. Robots can
mitigate this depletion by limiting their oxygen consumption, either overall
or in specific locations or situations.

##  1 Introduction

Implanted medical devices offer a wide range of benefits. Smaller devices will
enable additional high-precision applications [21, 61, 71, 73]. For example,
microscopic devices can act on specific cells based on logical combinations of
surface properties [49]. The development of autonomous microscopic robots [29,
43, 59, 61] could enable more complex selection criteria based on the local
environment of each robot. Autonomous robots determine when and how to use
harvested energy for tasks such as locomotion, communication or drug release
[11]. These decisions can account for sensed data, information received from
neighboring robots, and the history of such information saved in the robot’s
memory. Even relatively simple on-board control can provide a wide range of
customized behaviors [83]. For example, nearby robots could release drugs
simultaneously at a microscopic target location they identify with their
sensors [86] to produce a large sudden increase in drug concentration at that
location.

Realizing the potential of autonomous microscopic robots faces significant
challenges for fabrication and operation. One of these is power, which could
come from extending approaches used for larger implanted devices [1, 8, 19].
For tasks of limited duration, an onboard fuel source might suffice.
Otherwise, the robots need to obtain energy from their environment. One
demonstrated method to power microscopic devices is using external sources,
such as magnetic fields [54, 13] or ultrasound [32, 82]. Another approach is
chemical power [29], which is available throughout the body so robots do not
need to be close enough to the skin to receive adequate power from external
sources. Chemical power is especially useful for long-term tasks where it may
not be convenient or feasible to deliver power from outside the body or
provide implanted batteries. One possibility for chemical power is using
oxygen in the bloodstream [7, 20], such as by reacting glucose with oxygen [2,
15, 33, 67, 89]. In this case, oxygen is the limiting chemical because its
concentration is much lower than that of glucose [40]. Thus this paper focuses
on the availability of oxygen.

Of particular medical importance is the eventual development of autonomous
robots a few microns in size. These are the largest robots that can travel
throughout the entire circulatory system [29]. Micron-size robots are
considerably smaller than sub-millimeter micromachines [11] based on
microelectromechanical systems (MEMS). On the other hand, they are much larger
than nanoparticles used for drug delivery, which can only incorporate a few
logic operations [22]. Between these extremes, engineered biological cells
[53] can combine sensing, communication and logic processing in micron-size
devices. Theoretical studies suggest nonbiological micron-size robots
considered in this paper could have sufficient computation, sensing and
communication to coordinate complex activities with single-cell resolution and
a wider range of material properties and capabilities than biological cells
[29, 37, 51, 59, 75].

Microscopic robot applications can involve large numbers of robots. For
instance, micron-size robots could provide customized medical diagnostics and
treatments to individual cells throughout the body [43]. This task requires a
correspondingly large number of robots to provide a reasonable treatment time,
e.g., a few hours. Thus the scenarios discussed in this paper consist of
billions of robots, with a total mass in the range of tens to hundreds of
milligrams [29].

Interacting robots can create swarms with performance and robustness beyond
that of individual robots [26], including a variety of coherent structures and
behaviors [81]. For microscopic robots, external fields can induce these
interactions [58] to form, for instance, specific shapes [88] and fluid
motions [72, 36]. For more flexibility, interactions among autonomous robots
can arise by exchanging information with neighbors [70, 10].

Previous studies of chemical power for microscopic robots focus on one or a
few robots [29, 40]. Such studies are sufficient to evaluate available power
when the number of robots is too small to significantly affect oxygen
concentration on large scales. However, large numbers of robots could have
systemic effects in addition to local effects on nearby cells and robots. In
particular, robots powered by oxygen could significantly reduce oxygen
concentration in the blood before it reaches capillaries where blood delivers
oxygen to tissues.

Figure 1: Schematic illustration of a 2 $\mu$m-diameter spherical robot next
to a red blood cell.

This paper investigates how large numbers of robots small enough to fit
through capillaries and illustrated in Fig. 1 affect oxygen concentration as
they travel with the blood. Specifically, Section 2 describes a model of robot
transit and oxygen consumption. The model’s estimate for robot power and
oxygen concentration is described in Section 3 for a range of robot numbers.
In some cases, robots can significantly deplete oxygen in the blood. To avoid
this, Section 4 presents several mitigation strategies robots could use.
Section 5 describes limitations and extensions to the model that could cover a
wider variety of situations than considered here. Finally, Section 6
summarizes the model results and its implications for using chemical power for
microscopic robots.

##  2 Determining the Power Available to Circulating Robots

The model developed here for robot oxygen consumption consists of two parts:
the first evaluates how oxygen concentration changes during a typical
circulation loop, and the second determines how much power robots produce from
the available oxygen. Most of the circulation through the body passes through
a single capillary network between arteries and veins. We focus on this
typical case. Other situations, such as portal flows where the blood flows
through two sets of capillaries, require generalizing the model considered
here.

###  2.1 Robot Power Generating Capacity

We consider robots using fuel cells that oxidize glucose. Oxygen is the rate-
limiting chemical for this reaction since its concentration in the blood is
much lower than that of glucose. Thus the maximum power is when robots use all
oxygen reaching their surfaces. This is the case considered here, which
corresponds to the “high capacity robot with pump” design of Ref. [40].

When the distance between neighboring robots is large compared to their size,
a good approximation of robot oxygen collection is that of an isolated
absorbing sphere. Such a robot in blood plasma with oxygen concentration $c$
absorbs oxygen at the rate [9]

$J_{\mbox{\scriptsize robot}}=4\pi
D_{\textnormal{$\mbox{O}_{2}$}}r_{\textnormal{robot}}c$ (1)

where $D_{\textnormal{$\mbox{O}_{2}$}}$ is oxygen’s diffusion coefficient in
plasma and $r_{\textnormal{robot}}$ is the robot radius. Absorption can be
close to this value even when only a few percent of the robot surface absorbs
oxygen [9].

The reaction energy of glucose oxidation and rate of oxygen absorption
determine the power generated by the fuel cell. Not all of that power is
available for robot operations due to losses such as dissipation in wires
connecting fuel cells to loads in the robot and internal losses in the fuel
cell, e.g., due to membrane transport and incomplete catalysis. In addition,
robots could use pumps to improve oxygen collection, in which case the power
used by those pumps contributes a small amount the power system losses [29].
We characterize these losses by the fuel cell efficiency,
$f_{\textnormal{robot}}$: the fraction of the full reaction energy available
to the robot. Thus the power available to the robot from glucose oxidation is

$P_{\textnormal{robot}}=f_{\textnormal{robot}}\frac{e}{6}J_{\mbox{\scriptsize
robot}}$ (2)

where $e$ is the energy released by oxidizing a glucose molecule and the
factor of $1/6$ arises from the six oxygen molecules needed to oxidize each
glucose molecule.

###  2.2 Modeling Robot Oxygen Consumption

A small number of robots in the bloodstream have no systemic effect on oxygen
concentration. In that case, the normal concentrations found in the
circulation determine the power available to the robots and nearby tissue,
whether the robots are widely separated or form aggregates [40]. However,
large numbers of robots may consume a significant fraction of the oxygen in
the blood.

Red blood cells carry most of the oxygen in blood. These cells release oxygen
in partial compensation for that consumed by the robots. This replenishment
depends on the concentration of red blood cells, i.e., the hematocrit. In
small vessels, cells typically travel a bit faster than plasma, leading to a
reduced hematocrit in these vessels [57]. In addition, tissue consumes oxygen
from nearby capillaries.

Figure 2: Circulation from lungs to the rest of the body and back to the
lungs.

Estimating systemic effects of robot oxygen consumption does not require
precise vessel geometry of the many circulatory paths through the body.
Instead, we aggregate vessels of each size and consider flow through an
average aggregated structure for a single loop around the circulation shown in
Fig. 2. This loop starts as fully oxygenated blood leaves a lung capillary and
continues until the blood next reaches a lung capillary. A useful
simplification for evaluating the typical power available to robots is to
average over the vessel cross section, which gives a one-dimensional model of
the blood flow.

A circulation loop takes about one minute. Evaluating behavior on this time
scale averages over the short term variations in flow speeds, particularly in
arteries, due to heart beats. We also consider a fixed, resting pose for the
body and do not treat longer-term variations from changes in activity level or
environmental factors. This temporal averaging leads to a steady-state profile
of oxygen concentration in a typical circulation loop. Appendix A provides the
details of this vessel model, which combines spatial aggregation and temporal
averaging. This aggregated model contrasts with more detailed models of small
portions of vessels required to evaluate the local effects of a small number
of robots [40].

Figure 3: Compartment model of oxygen flows after blood leaves the lungs.

Consider a small volume of blood moving through the circulation, containing
plasma, cells and robots. The oxygen concentration model consists of four
compartments. Blood vessels contain three of these compartments: blood plasma,
blood cells, and robots. The robots occupy a small fraction of the plasma
volume. The fourth model compartment is the tissue around capillaries. Oxygen
concentration in the moving volume of blood changes due to consumption by
robots and tissue, and oxygen release by red blood cells. Fig. 3 illustrates
these processes, with details given in Appendix C.

##  3 Robot Power and Oxygen Concentration

|
---|---
(a) | (b)
Figure 4: Schematic illustration of typical spacing between robots, indicated as points, in vessels. The arrows indicates the neighbor closest to the robot in the center of each diagram. (a) In large vessels, the nearest robots are within the same vessel. (b) In small vessels, the nearest robot may be in a neighboring vessel. |  | robot | robot spacing
---|---|---|---
number | nanocrit | number density | large vessels | capilliaries | body
$10^{10}$ | $8\times 10^{-6}$ | $2\times 10^{12}/\mbox{m}^{3}$ | $80\,\mbox{$\mu$m}$ | N/A | $170\,\mbox{$\mu$m}$
$10^{11}$ | $8\times 10^{-5}$ | $2\times 10^{13}/\mbox{m}^{3}$ | $40\,\mbox{$\mu$m}$ | N/A | $80\,\mbox{$\mu$m}$
$10^{12}$ | $8\times 10^{-4}$ | $2\times 10^{14}/\mbox{m}^{3}$ | $20\,\mbox{$\mu$m}$ | $100\,\mbox{$\mu$m}$ | $40\,\mbox{$\mu$m}$
Table 1: Properties of various numbers of robots in the circulation. Nanocrit
is the fraction of the blood volume occupied by robots, in analogy with
hematocrit, which is the fraction occupied by red blood cells. The robot
spacing is typical distance between neighboring robots. The value for large
vessels is the cube root of the average blood volume per robot. In
capillaries, the spacing is from the blood volume in a
$8\,\mbox{$\mu$m}$-diameter vessel, up to a maximum capillary length of
$1\,\mbox{mm}$. The spacing in the body is the cube root of the average body
volume per robot for a nominal $50\,\mbox{L}$ body volume.

This section applies the model described in Section 2 to determine robot power
and oxygen concentration for the robot numbers given in Table 1, which are
typical values for proposed applications [29]. The largest number of robots
considered here, $10^{12}$, have a combined mass of several grams and volume
of a few milliliters. The table gives typical spacings between robots in large
vessels, within capillaries and between nearby small vessels, as illustrated
in Fig. 4. These spacings are much larger than the size of the robots so they
do not directly compete with neighboring robots for oxygen, in contrast to
situations where robots operate in close proximity [40].

Micron-size robots have considerably smaller volume than red blood cells, so
even the largest number of robots considered here occupy less than $0.1\%$ of
the blood volume, compared to 45% typically occupied by blood cells. This
small fraction of robots does not significantly affect blood rheology [29],
allowing the model to use typical flow speeds in the absence of robots.

###  3.1 Power when Robots Maximally Consume Oxygen

Figure 5: Power used by a robot during a circulation loop: starting when a
robot leaves a lung capillary and ending just before it next enters a lung
capillary. The light vertical lines indicate when the blood is in a capillary,
where tissue consumes oxygen from the blood.

Fig. 5 shows how robot power varies during a circulation loop when robots
consume oxygen as fast as it diffuses to their surfaces, so robot power is
given by Eq. 2. Robot power decreases as the robot moves from the lung:
hundreds of picowatts in the arteries, an abrupt reduction in the capillary
where tissue competes with the robots for oxygen, and a continued gradual
decrease as the robot travels through veins back to the lung.

The consequences of this variation in power during a circuit depend on how
well it matches the power requirements of the robots’ tasks. For instance, if
the robots require some power to maintain their activity, the minimum power
available during the circulation, i.e., just before returning to the lung, is
the relevant model result. If this minimum is below the required power, either
the task will need to use fewer robots or the robots will need sufficient
onboard energy storage to support their activities until they return to the
lung.

Some applications have flexible timing of robot power demand. Such tasks could
include computation to evaluate sensor measurements, maintenance tasks such as
checking robot functionality, and communicating information to other robots.
In such cases, robots could wait until there is abundant power to perform
these tasks, e.g., while passing through arteries.

In other cases, the robots might have highest power demand during the short
time they pass through capillaries. For instance, robots might need to move to
cells near the capillary that are emitting specific chemicals into the blood,
thereby requiring the robots to measure, compute and propel themselves while
in the capillaries [37]. In this case, the relevant value from Fig. 5 is the
power available while robots pass through capillaries.

###  3.2 Consequences of Robot Oxygen Consumption

|
---|---
(a) | (b)
Figure 6: (a) Oxygen concentration in plasma expressed in molecules per unit
volume. For its relation to units more commonly used in macroscopic settings,
see Appendix D. (b) Oxygen saturation of red cells. The light vertical lines
indicate when the blood is in a capillary.

For further insight into the consequences of robots using oxygen for power,
Fig. 6 shows oxygen concentration in the plasma and red cell saturation. The
figure compares these values with those without robots, in which case the only
oxygen consumption occurs in tissue around the capillaries.

Except for the largest number of robots considered here, the diffusion limit
on the rate oxygen reaches the robot surface (i.e., Eq. 1) prevents the robots
from fully depleting oxygen in the blood. That is, as is the case without
robots, the blood returns to the lungs with much of the oxygen it originally
took from them. By contrast, a trillion robots consuming oxygen as fast as
possible completely depletes oxygen by the end of the circuit.

####  3.2.1 tissue

Figure 7: Relative oxygen concentration around an absorbing robot next to a
blood cell, ranging from 1 at the cell surface to 0 at the robot. Figure 8:
Tissue power relative to its maximum value with unlimited oxygen.

An important consequence of robots consuming oxygen is their effect on oxygen
available to tissue. Robots consume oxygen in arteries bringing oxygen to
capillaries and compete with tissue for oxygen in the capillaries. Much of the
oxygen robots use comes from red cells because absorbing robots produce large
concentration gradients at nearby red cells, as illustrated in Fig. 7. This
additional oxygen from red cells limits the decrease in concentration in
plasma. Thus robot consumption leads to only modest reduction of oxygen in the
capillaries, as shown in Fig. 6a.

The extent to which this reduced concentration affects tissue depends on the
minimum oxygen tissue requires to support its metabolism. Tissue demand leads
to the abrupt reduction in concentration in capillaries seen in Fig. 6a. Even
with the largest number of robots considered here, the concentration in
capillaries is sufficient to support resting tissue demand. Specifically,
cellular functions continue to operate normally until oxygen partial pressure
is below $5\,\mbox{mmHg}$ [23], corresponding to concentration $0.4\times
10^{22}\,\mbox{molecule}/\mbox{m}^{3}$. This is lower than the capillary
concentrations seen in Fig. 6a.

The conclusion of sufficient oxygen for tissue also follows from the tissue
power generation described by Eq. C.3. With the tissue parameters in Table
D.1, tissue power is nearly independent of oxygen concentration at the
concentrations seen in Fig. 6a. Quantitatively, Fig. 8 shows the reduction in
oxygen in capillaries due to robots has only a minor effect on tissue power in
the case considered here, i.e., resting metabolic demand. Thus blood typically
transports much more oxygen than required by tissue and so provides a buffer
for any localized increase in tissue metabolic demands.

####  3.2.2 vein walls

Fig. 6 indicates the lowest oxygen concentrations occur toward the end of the
circulation loop, as robots return to the heart and then to the lungs.
Normally, blood returning to the lungs contains a significant portion of the
oxygen originally collected in the lungs. Robots can make use of this oxygen
without concern of reducing oxygen available to tissue since this blood has
already passed through capillaries to deliver oxygen to tissue.

A caveat to this conclusion is that cells comprising the walls of blood
vessels consume a portion of the oxygen carried in those vessels.
Specifically, small arteries and veins, and the inner portion of walls of
larger vessels, receive nutrients that diffuse into the wall from the blood
carried in the vessel [69]. These cells form a small fraction of the body
tissue so their oxygen use is not a significant contribution to total tissue
oxygen consumption. Moreover the small diffusion distance of oxygen and the
relatively large volume of the vessels compared to capillaries means
consumption by the vessel walls does not significantly alter the concentration
in the vessels during the time blood flows through them. Thus there is no need
to include this oxygen consumption in the model.

However, the lowered oxygen concentration in veins due to the robots could be
a safety issue. For example, normal leg veins have oxygen concentrations of
about $30\text{--}40\,\mbox{mmHg}$ and red cell saturation $50\text{--}70\%$
[52]. As discussed in Appendix D, $30\,\mbox{mmHg}$ is $2.5\times
10^{22}/\mbox{m}^{3}$. Thus these ranges are a bit below the values toward the
end of the circulation loop with $10^{11}$ robots. This comparison suggests
lowered concentration in veins will not be an issue when using that many
robots, but could be important for larger numbers of robots. Thus the oxygen
requirements of vein walls may place more stringent constraints on robot
oxygen consumption than the requirements of most tissue, which receives
nutrients from capillaries. Quantifying this constraint requires determining
the minimum concentration these cells can tolerate for various amounts of
time. An initial assessment is to assume these cells have requirements similar
to those of resting tissue discussed above. In that case, Fig. 6a shows that
$10^{12}$ robots could be detrimental to vessel walls over the last
$15\,\mbox{s}$ or so of the circulation.

####  3.2.3 heating

number | average power | minimum power | total dissipation
---|---|---|---
$10^{10}$ | $460\,\mbox{pW}$ | $380\,\mbox{pW}$ | $9\,\mbox{W}$
$10^{11}$ | $325\,\mbox{pW}$ | $240\,\mbox{pW}$ | $65\,\mbox{W}$
$10^{12}$ | $120\,\mbox{pW}$ | none | $240\,\mbox{W}$
Table 2: Power for robots during circulation: average over the circulation,
minimum (just before returning to the lungs), and total dissipation from the
oxygen consumed by all the robots.

Robot power production adds heat to the body. This heating arises from the
full reaction energy in the fuel cells, not just the fraction available for
the robot’s use. Thus the total dissipation from all the robots equals the
average available power over a circulation, multiplied by the number of robots
and divided by the fuel cell efficiency. Table 2 gives these values for the
numbers of robots in Table 1. For comparison, typical basal metabolism is
$100\,\mbox{W}$ or $2000\,\mbox{kcal}/\mbox{day}$.

The table shows $10^{10}$ robots, each consuming as much oxygen as it can, add
less than $10\%$ to basal metabolism. On the other hand, $10^{12}$ robots add
more than twice the basal amount, thereby adding significant heat to the body.
This is well below the heat dissipation during exercise [29], but nevertheless
may be a limiting factor during prolonged robot operation.

####  3.2.4 oxygen concentration measurement

The oxygen concentration profile produced by robots alters assumptions
underlying some clinical measurements, particularly the relation between
direct measurements and inferred properties based on causal relationships in
the body [6]. For instance, normally oxygen is only removed from blood in
capillaries so the arterial concentration is the same throughout the body.
This is the basis for pulse oximetry, a common, noninvasive measurement of
respiratory heath and oxygen supply to organs [62].

Robots consume oxygen in arteries, leading to significant concentration
reduction in the vessels between the lung and capillaries, as shown in Fig. 6.
Thus in the presence of robots, pulse oximetry will require modified
calibration to account for variation in arterial oxygen depending on body
location and the time since the blood left the lung. These systemic changes
could also affect the interpretation of other large-scale functional oxygen
measurement techniques, such as photoacoustic imaging [12]. The model
described here indicates how oxygen varies with location and could aid in
calibrating and interpreting these measurements.

As compensation for their alteration of conventional clinical measurements,
robots will be able to measure oxygen concentration throughout the body at
micron length scales, thereby far surpassing the accuracy and quantity of
external measurements. Interpreting such measurements will need to account for
the systemic changes in concentration produced by the robots. In spite of this
robot capability, recalibrated conventional sensors will remain useful checks
on robot performance and allow comparison with established clinical practice.
Moreover, receiving measurements from robots may be delayed due to
communication limits, e.g., if robots must wait until they are near the skin
to send information out of the body. Delays or reduced information could also
occur if continual communication takes too much power away from the robots’
main tasks. In such cases, the improved sensing capability of robots will not
be as readily available as measurements from conventional sensors.

###  3.3 Robots Operating Only in Capillaries

|
---|---
(a) | (b)
Figure 9: Behavior with all robots in capillaries as a function of distance
along a capillary, with arterial flow arriving from the left. (a) Robot power.
(b) Oxygen concentration in plasma.

The focus of this paper is on robots generating power as they travel with the
blood. In some applications, robots perform most of their tasks in
capillaries, e.g., to monitor or act on individual cells accessed from
capillaries. In this case, robots could initially travel through the
circulation until they reach capillaries, at which point they attach to the
capillary walls to perform their primary function. Robots in lung capillaries
would have access to the oxygen not taken by red cells, as discussed in
Section 4.1. However, robots in capillaries of the systemic circulation would
only consume oxygen from blood passing through capillaries, rather than during
the entire circulation.

Applying the model described in Section 2 to robots positioned in systemic
capillaries requires two modifications. First, there is no robot oxygen
consumption in arteries or veins. Second, since these capillaries contain
about $5\%$ of the total blood volume [29], the number density of robots in
capillaries is about 20 times larger than when they are distributed throughout
the blood volume. With these modifications, Fig. 9 illustrates how robot power
and oxygen concentration vary in a capillary with typical length of
$1\,\mbox{mm}$. Since there is no oxygen consumption in arteries, the
concentration in blood entering the capillary does not depend on the number of
robots. Oxygen concentration and robot power decrease through the capillary,
so robots near the venous end of the capillary have less power. If necessary,
robots near the arterial end of the capillary could reduce their power to
leave more available for downstream robots, as a small-scale version of the
mitigation strategy discussed in Section 4.2.

Comparing with Fig. 6a shows that the oxygen concentration in blood after
passing through a capillary is similar to that when robots are distributed
throughout the blood and consume all oxygen reaching their surface. That is,
in both cases, the robots extract about the same amount of oxygen from
circulating blood, up to the time it passes through a capillary. Thus the
average power per robot and their total dissipation are similar to the values
in Table 2. However, this power is entirely generated in capillaries, giving a
much larger power density in the blood as it passes through capillaries. For
example, $10^{10}$ robots heat capillaries at an average rate of about
$30\,\mbox{kW}/\mbox{m}^{3}$. This is several times the resting tissue power
demand used in this model (see Table D.1), and comparable to the power density
for cells with high metabolic demands, such as heart or kidney cells [29].
Nevertheless, heat transport by blood and through tissue is rapid enough at
these small scales that this increased power density does not result in much
local temperature increase, even when robots are close enough to come into
contact [40]. This indicates that even when all robots operate in capillaries,
the main heating issue is for the body as a whole, as discussed in Section
3.2.3, rather than local heating in capillaries.

Fig. 9 shows the average behavior in capillaries. However, the small blood
volume in individual capillaries leads to considerable variation due to
differences in capillary type [29], flows within a network of connected
capillaries and the precise locations of robots in a capillary. For instance,
at the lower range of the numbers of robots considered here, each capillary
will have only a few robots, on average, so that the actual number of robots
will vary considerably among capillaries, including many capillaries with no
robots.

Figure 10: Five robots anchored to the inner wall of an
$8\,\mbox{$\mu$m}$-diameter vessel with $5.5\,\mbox{$\mu$m}$ spacing along the
vessel. The vessel segment shown here is $34\,\mbox{$\mu$m}$ long and is shown
with a cross section through the middle of the vessels and robots. Successive
robots positioned on opposite sides (top) and the same side (bottom) of the
vessel. Figure 11: Relative oxygen concentration on a vertical slice through
the center of the vessel and the robots, ranging from one at the inlet at the
left to zero at the surface of each robot due to its absorption of all oxygen
reaching it. The white curves are streamlines of the oxygen flux. The rows
correspond to the robot positions shown in Fig. 10. The fluid flows from left
to right in each diagram, with average speed $0.2\,\mbox{mm}/\mbox{s}$ and
$1\,\mbox{mm}/\mbox{s}$ in the left and right columns, respectively. The
oxygen diffusion coefficient is in Table D.1.

The circulation model considered in this paper does not evaluate variation in
oxygen concentration at micron length scales. For example, it does not
consider oxygen diffusion across the width of a vessel. Moreover, as discussed
in Appendix C, the model assumes robots are sufficiently far apart that they
do not compete with neighboring robots for oxygen. From Table 1, this is
reasonable even when all the robots are in capillaries, up to about $10^{11}$
robots or so. However, with $10^{12}$ robots in capillaries, the distance
between neighboring robots is only somewhat larger than their diameter. Thus,
that many robots will have some competition for oxygen, reducing the oxygen
they collect and the power they generate compared with the model’s
predictions.

To illustrate the effect of competition among neighboring robots and diffusion
across capillaries, Fig. 10 shows two examples of robots in a small segment of
a capillary with the average spacing corresponding to $10^{12}$ robots in the
capillaries. These examples differ in the positioning of successive robots:
alternating between opposites sides of the vessel or all on the same side.
Fig. 11 shows the resulting oxygen concentration for four scenarios: the two
robot positions shown in Fig. 10 and with two fluid flow speeds that are a
typical range for capillaries [29]. This figure shows large variations in
concentration along the vessel, particularly near each of the robots. This
contrasts with the monotonically decreasing concentration from the averaged
circulation model shown in Fig. 9b.

The concentration variation across the vessel is particularly large at the
higher flow speed, where oxygen does not have enough time to diffuse
throughout the vessel cross section while the fluid passes the robots. Robots
positioned on the same side of the vessel increase the difference in oxygen
concentration between the two sides of the vessel compared to robots on
alternate sides. The differences in oxygen concentration around the robots
leads to variation in the rate oxygen molecules reach the robot surface
depending on robot position and fluid speed, as shown by the oxygen flux
streamlines in Fig. 11.

While the large-scale model shown in Fig. 9 describes behavior averaged over
capillaries, the oxygen concentration and robot power will vary considerably
from this average in individual capillaries. This is due to variations in the
number of robots in the small volume of blood in a capillary and the positions
of the robots on the capillary walls. Additional variations could arise from
features of the flow not included in this small-scale model of oxygen
transport in capillaries. For instance, the model neglects oxygen consumption
by nearby tissue cells and the possibility that some oxygen reaching nearby
tissue diffuses back to the robots in the vessel. Moreover, the model assumes
the oxygen saturation of passing blood cells remains in equilibrium with the
concentration in the fluid. Accounting for tissue consumption, diffusion
outside the vessel, and the kinetics of oxygen release from blood cells gives
similar large concentration gradients near absorbing robots on a vessel wall
[40]. Additional changes to the oxygen concentration could arise from blood
cells as they distort to move past robots anchored to the vessel wall.

An additional consideration for robots remaining in capillaries rather than
moving with the blood is the increase in vascular resistance of the flow,
particularly in the narrowest capillaries since vascular resistance of these
vessels is inversely proportional to the fourth power of their diameter
according to Poiseuille’s law [35]. For example, comparing the pressure drop
for a vessel without robots to that of vessels with robots indicates the
robots increase vascular resistance by about $20\%$ in both configurations
shown in Fig. 10. Robots on vessel walls may further increase the resistance
by changing how blood cells distort as they move past the robots. As an
extreme, robots could block passage of the cells.

If this increased resistance reduces the blood flow, robots and the tissue
around the capillaries will not receive new oxygenated blood as rapidly as
assumed by the circulation model. Alternatively, the body may compensate for
the increased resistance by a corresponding increase in the blood pressure.
Avoiding injury from robot blockage or increased blood pressure will limit the
number of robots that can be stationed in capillaries, and how long they can
remain there. This could be an important limitation on applications that
require longitudinal cell-specific data on many cells over an extended period
of time.

An alternative way to collect information from the same cell over a period of
time that avoids blocking capillary blood flow is using robots that move with
the blood, as considered by the average circulation model of this paper. The
robots collect data while passing through a capillary. By collecting
sufficient data to uniquely identify the capillary and their location within
it, post-processing could match data collected by different robots, at
different times, from the same capillary, thereby reconstructing longitudinal
observations with single-cell resolution. Instead of continuously monitoring
cells, this would collect snapshots each time a robot passes through a
capillary near the cell.

On average, each of $n$ robots completes a circulation in $t=60\,\mbox{s}$ and
passes through about $1.25$ capillaries in the systemic flow (including a
portion portal flows that pass through more than one capillary during a single
circuit [25]). Thus, on average, robots pass through a given capillary at the
rate $r=1.25n/t/N$ where $N\approx 2\times 10^{10}$ is the number of
capillaries in the systemic circulation [29]. The average time between
successive robots is $1/r$. For example, with $n=10^{12}$ robots, a robot
passes through a capillary about once a second, though with considerable
variation around this average value. If this is adequate temporal resolution
for the task, and robots can collect data while moving with the flow instead
of, e.g., requiring probes into the vessel wall, then collecting data as
robots move through capillaries is a viable alternative to robots attached to
the wall. In either case, the robots would mainly be active, and consuming
oxygen, during the time they are in capillaries, so Fig. 9 describes their
average power. This alternative places larger demands on robot data storage
since they will need to collect not only the cell measurements of interest but
also the data required provide the unique identifications. On the other hand,
robots moving in capillary networks could also use their interactions with the
flow and other robots to perform distributed microfluidic computation [65] to
provide useful information on the microcirculation.

###  3.4 Anemic Patients

|
---|---
(a) | (b)
Figure 12: Behavior with 25% hematocrit. (a) Robot power. (b) Oxygen
concentration in plasma. The light vertical lines indicate when the blood is
in a capillary.

Disease and injury could alter the amount of oxygen available to robots. For
example, patients with anemia, i.e., having lower than normal hematocrit, have
less ability to replenish the oxygen robots remove from plasma than people
with normal hematocrit.

Fig. 5 shows power available to robots in an individual with normal
hematocrit. This model can illustrate the effect of anemia by reducing the
hematocrit parameter. As an example, Fig. 12 shows behavior with $25\%$
overall hematocrit. Even with this lowered hematocrit, the available oxygen
from circulating cells provides significant power for all but the largest
number of robots considered here. However, $10^{12}$ robots remove all oxygen
before blood reaches capillaries. Thus for the lower range of robot number,
anemia does not significantly change the situation for robots using chemical
power. But the low oxygen reserve in cells with this reduced hematocrit
significantly increases oxygen depletion with a large number of robots.

This example shows how the model discussed in this paper can evaluate the
suitability of patients for applications of microscopic robots. Personalized
versions of such models could help develop mission plans for the robots and
provide a baseline to compare with robot measurements during early stages of a
mission to identify and respond to deviations from the plan before they become
harmful. This extends to microscopic robots the current use of computational
models to help plan conventional medical procedures [79].

As described in Section 3.2.4, conventional monitoring may have reduced
accuracy when large numbers of robots are consuming oxygen. The variation in
response to large numbers of robots based on the patient’s health status
suggests a staged approach of gradually increasing the number of robots in the
body. Another way to achieve a similar effect is by having robots initially
limit their power use and gradually increase that limit to the level required
for their mission. During this initial stage, the robots could monitor their
effect on oxygen concentration and other body processes to determine whether
full power would exceed safety limits prior to fully activating all the
robots. This evaluation would allow determining patient-specific trade-offs in
treatment options. For example, a treatment could use fewer robots or have
them operate at lower power with the trade-off of longer treatment duration or
less frequent communication with the robots. In particular, monitoring body
function during a mission would allow adapting robot behavior and mission
parameters to far more detailed measurements than would be available from
conventional macroscopic sensors. Continual comparison with personalized
versions of the model discussed here could aid in the interpretation of these
measurements and anticipate the effect of increasing the number of robots or
their power use.

##  4 Mitigating the Effects of Robot Oxygen Consumption

As described in Section 3, $10^{12}$ robots lead to significant oxygen
reduction and heating when they consume oxygen as fast as possible. Thus the
application of microscopic robots could be limited to fewer robots, a shorter
mission duration or intermittent operation over a longer period of time. If
these adjustments significantly degrade mission performance, altering robot
design or behavior are other approaches to mitigate these effects. This
section discusses such approaches, which involve either enhancing the oxygen
carried in the blood or robots consuming less of the oxygen available to them.

###  4.1 Storing Oxygen

Normally, oxygen collection in the lungs is perfusion limited [25], i.e.,
oxygen in lung capillaries is more than sufficient to fully saturate blood
plasma and red cells. Robots with storage tanks could collect some of this
excess oxygen without reducing oxygen collected by red blood cells, thereby
increasing the total circulating oxygen. This section discusses several ways
stored oxygen could supplement the oxygen available to the robots, with the
caveat that the amount of additional oxygen depends on the health of the
lungs: patients with limited lung capacity may not have sufficient additional
available oxygen for robots to collect.

Robots could fill tanks from oxygen in vessels other than lung capillaries.
However, this would reduce the oxygen available later in the circulation in
the same way as robots immediately using all oxygen reaching their surfaces,
as shown in Fig. 6a. Thus, the discussion in this section focuses on robots
obtaining oxygen in lung capillaries.

####  4.1.1 oxygen storage tanks

Figure 13: Cross section of an oxygen transport robot with volume fraction $f$
devoted to oxygen storage (blue) and storage tank wall thickness $t$. For
clarity, the wall thickness is exaggerated compared to the other dimensions.

A robot stores oxygen in a pressure tank [29]. Fig. 13 illustrates a robot
with a spherical oxygen tank with wall thickness $t$ and interior volume that
is a fraction $f$ of the robot’s volume.

The amount of oxygen a robot can store is limited by three factors: the rate
at which a robot absorbs oxygen, the time it spends in a lung capillary, and
its storage capacity.

##### Absorption

Oxygen absorption depends on how fast molecules diffuse to the robot’s surface
and the fraction of the surface that captures arriving oxygen with pumps. Eq.
1 gives the rate oxygen diffuses to the robot surface. The corresponding flux
to the surface is $J_{\mbox{\scriptsize robot}}/(4\pi
r_{\textnormal{robot}}^{2})$. For a $1\,\mbox{$\mu$m}$-radius robot
$J_{\mbox{\scriptsize robot}}=1.8\times 10^{9}\,\mbox{molecule}/\mbox{s}$ in
lung capillaries, corresponding to a flux of $1.4\times
10^{20}\,\mbox{molecule}/\mbox{m}^{2}/\mbox{s}$.

Figure 14: Fraction of robot surface covered with pumps to absorb oxygen in
lung capillaries as a function of robot radius. In the shaded area on the
left, molecules arrive too fast for pumps to collect, even if they cover the
entire surface. In the shaded area on the right, surface coverage is $5\%$,
which is sufficient for pumps to capture most of molecules reaching the robot
surface. For intermediate sizes, the surface fraction is from Eq. 3.

For maximum absorption, the pumps must have sufficient capacity to collect
molecules as fast as they reach a pump. However, pumps have a maximum
operating rate $J_{\mbox{\scriptsize pump}}=(4\pi
r_{\textnormal{robot}}^{2})sF_{\mbox{\scriptsize pump}}$ where $s$ is the
fraction of the robot surface used by pumps and $F_{\mbox{\scriptsize pump}}$
is their maximum pump capacity. For molecular sorting rotors, a plausible
capacity is $F_{\mbox{\scriptsize
pump}}=10^{22}\,\mbox{molecule}/\mbox{m}^{2}/\mbox{s}$ [29]. Capturing all the
molecules requires that $J_{\mbox{\scriptsize pump}}\geq J_{\mbox{\scriptsize
robot}}$. With Eq. 1, the minimum surface coverage is

$s=\frac{D_{\textnormal{$\mbox{O}_{2}$}}c}{F_{\mbox{\scriptsize
pump}}}\frac{1}{r_{\textnormal{robot}}}$ (3)

with $c$ the oxygen concentration in lung capillaries.

Another constraint on pump surface fraction arises from diffusion. When pumps
do not cover the entire surface, an oxygen molecule diffusing to the robot’s
surface will not necessarily reach a pump. Nevertheless, once a molecule
reaches any part of the robot’s surface, it usually moves for a considerable
time near the surface before diffusing away. This means that pumps that only
cover a few percent of the surface can collect oxygen nearly as well as a
completely absorbing surface [9]. This provides a lower limit on the fraction
of the surface used by pumps. An additional constraint arises from the
multiple uses a robot has for its surface, e.g., for sensing and structural
support. Thus pumps cannot occupy the entire surface.

These three constraints (pump capacity, diffusion rate to the robot surface
and fraction of the surface available for pumps) combine to determine the rate
robots can absorb oxygen. Fig. 14 illustrates how these constraints determine
the fraction of surface area needed for pumps to absorb most of the arriving
oxygen when the robot is in a lung capillary as a function of robot size. This
example uses a minimum fraction of $5\%$, which allows capturing most arriving
oxygen [9]. At small sizes, even if pumps cover the entire surface, they
cannot absorb all the oxygen reaching the surface. However, even if the other
components a robot needs on its surface limit pumps to no more than $25\%$ of
the surface, Fig. 14 shows that the limit due to pump capacity only applies to
robots significantly smaller than used in the scenarios of this section.

##### Time

Blood passes through a lung capillary in about $0.75\,\mbox{s}$ [25]. We use
this as the oxygen collection time for robots moving with the blood, although
robots might have somewhat less time as they move with cells, due to the
reduced hematocrit in small vessels (see Appendix A.1). Robots could increase
the filling time in several ways. Robots moving passively with the blood could
use several circulations through the lungs to fill their tanks. Active robots
could extend their filling time by sticking to capillary walls or selecting
longer routes through the lung capillary network.

##### Capacity

The tank stores oxygen at high pressure. For a spherical tank of radius $r$
with a thin wall of thickness $t$, Laplace’s law gives its maximum pressure as
[29]

$p_{\textnormal{max}}=\frac{2t\sigma}{r}$ (4)

where $\sigma$ is the wall’s failure strength. For a wall formed of
covalently-bonded carbon, a conservative estimate of the failure strength is
$\sigma=10^{10}\,\mbox{Pa}$, about $20\%$ of diamond’s failure strength [29].
For example, a tank with radius $r=0.3\,\mbox{$\mu$m}$ and wall thickness
$t=5\,\mbox{nm}$ has maximum pressure near $3000\,\mbox{atm}$.

The scenarios discussed in this section use tanks storing oxygen at about one-
third of this maximum pressure. At these storage pressures and body
temperature, common gases such as oxygen deviate somewhat from the ideal gas
law. To account for this deviation, we estimate the tank storage capacity
using the van der Waals equation of state for oxygen [29].

####  4.1.2 diffusion-limited storage

During a $0.75\,\mbox{s}$ transit through a lung capillary [25], $1.3\times
10^{9}$ molecules diffuse to the surface of a $1\,\mbox{$\mu$m}$-radius robot.
For example, at body temperature and tank pressure of $1000\,\mbox{atm}$, the
number density of oxygen molecules is $1.26\times
10^{28}\,\mbox{molecule}/\mbox{m}^{3}$ [29], which is far larger than the
concentration in blood (see Table D.1). Thus a tank with radius
$0.3\,\mbox{$\mu$m}$ could store all the collected molecules. This tank
occupies about $3\%$ of the robot’s volume, including a $5\,\mbox{nm}$-thick
wall.

From Eq. 2, a robot collecting oxygen as fast as it diffuses to its surface
while passing through a lung capillary collects enough oxygen to provide about
$7\,\mbox{pW}$ for the duration of a $60\,\mbox{s}$ circulation loop. This is
well below the power available from oxygen in the blood during the circulation
with even as many as $10^{12}$ robots, except for the last ten seconds or so
of the loop (see Fig. 5). If the stored oxygen were only used during the last
ten seconds of the circulation, it would provide about $50\,\mbox{pW}$. Thus
this oxygen storage is not sufficient for robots requiring around
$100\,\mbox{pW}$ or so, and hence can not completely alleviate depletion by
large numbers of robots consuming oxygen. However, stored oxygen would be
useful as a supplemental power source for brief intervals of reduced oxygen,
such as when a robot is next to a white blood cell moving through a capillary,
so the plasma around the robot is temporarily not replenished by nearby red
blood cells.

####  4.1.3 tank capacity-limited storage

As an illustration of oxygen storage, suppose robots collect enough oxygen to
provide $100\,\mbox{pW}$ for the $60\,\mbox{s}$ duration of the average
circulation loop, which corresponds to $1.8\times 10^{10}$ oxygen molecules.
Storing this much oxygen in a pressure tank at one-third its maximum pressure
requires about one-third of the robot volume for a tank with $10\,\mbox{nm}$
thick walls. Filling this tank requires $10\,\mbox{s}$ in a lung capillary,
which is much longer than typical transit times. To fill its tank, a robot
could remain in a lung capillary rather than flow with the blood, e.g., by
anchoring itself to the capillary wall. Lung capillaries contain
$70\,\mbox{mL}$ of blood [25]. Thus if robots stay in capillaries long enough
to fill their tanks, lung capillaries would hold about $15\%$ of the robots in
only $1.2\%$ of the blood volume.

For $10^{12}$ robots, this scenario gives a nanocrit of $0.01$ in lung
capillaries, about ten times larger than the overall nanocrit (see Table 1).
The distance between robots in a capillary is correspondingly reduced, to
about $10\,\mbox{$\mu$m}$. This spacing is small enough that nearby robots
compete for oxygen, thereby somewhat increasing the filling time beyond that
determined here for isolated robots [40].

These estimates indicate the trade-offs required for robots to carry enough
oxygen for $100\,\mbox{pW}$ without consuming oxygen from the blood during the
circulation. In particular, this scenario requires a large fraction of robot
volume for storage, significantly increases robot concentration in lung
capillaries and requires robots have the capability to anchor themselves in
capillaries.

####  4.1.4 oxygen transport by specialized robots

An alternative to oxygen tanks in the robots performing the medical mission is
to use additional specialized oxygen transport robots [28]. These robots
collect oxygen while passing through the lungs and deliver it to the main
mission robots in the systemic circulation.111This oxygen delivery contrasts
with transport robots that also collect carbon dioxide for return to the lungs
[28]. Such robots require power to collect and compress molecules while in the
systemic circulation, which robots that only deliver oxygen do not need.

Oxygen transport robots face the same limit on extracting oxygen from lung
capillaries as discussed above, depending on their size and tank capacity.
However, the transport robots can be optimized to address these limitations
without compromising the main mission.

For example, in the scenario of Section 4.1.3, robots cannot fill their tanks
during a single transit through a lung capillary. Oxygen transport robots can
fill their tanks in this time by exploiting the geometry of diffusion: from
Eq. 1 the rate molecules arrive at the robot is proportional to its radius
$r_{\textnormal{robot}}$, while tank capacity is related to its volume, which
is proportional to $r_{\textnormal{robot}}^{3}$ when tanks use a fixed
fraction of the robot volume and a fixed storage pressure. Thus the time to
fill a tank scales as $r_{\textnormal{robot}}^{2}$. This means that instead of
a single large tank in one robot, using proportionally smaller tanks in many
robots, with the same overall storage capacity, can collect the oxygen in a
shorter time. Thus, for example, decreasing the robot size by a factor of two
reduces the time required to fill its tank by a factor of four. Increasing the
number of robots by a factor of eight gives the same total volume of robots
and the amount of oxygen they can transport.

To quantify the design trade-offs for these robots, consider $n$ oxygen
transport robots, each of radius $r$ with a fraction $f$ of its volume for
oxygen storage, as shown in Fig. 13. Scaling the entire geometry with robot
size means the thickness of the oxygen tank wall is proportional to $r$. From
Eq. 4, this scaling has the same maximum pressure for the storage tank in
robots of different sizes. Specifically, the wall thickness in this example is
$t=20\,\mbox{nm}\times(r/r_{\textnormal{robot}})$ where
$r_{\textnormal{robot}}=1\,\mbox{$\mu$m}$ is the radius of the main robots.
The wall cannot be less than one atomic layer, so this scaling does not apply
for arbitrarily small robots. E.g., with a minimum $t\geq 1\,\mbox{nm}$
thickness, this relation applies for $r_{\textnormal{robot}}\geq
0.05\,\mbox{$\mu$m}$. The transport robots considered here are larger than
this size. As with the example of Section 4.1.2, tanks store oxygen at one-
third of their maximum pressure.

With this specification of the transport robots, we select the number of
robots, their size and volume devoted to oxygen, i.e., values for the
parameters $n$, $r$ and $f$, so the robots, in aggregate, carry $1.8\times
10^{10}$ oxygen molecules for each of the main robots, duplicating the
scenario of Section 4.1.3. In addition to this requirement, we consider three
constraints on the choice of these parameters for transport robots.

First, the robots should be small enough so the favorable diffusion scaling
allows them to fill their tanks with a single transit of the lung capillaries.
This includes setting the fraction of the surface used by pumps as shown in
Fig. 14 so the robots can collect the oxygen available to them in the
capillary.

Second, the volume of the robot outside its oxygen tank must be sufficient to
contain its other components. These additional components include pumps to
collect oxygen and mechanisms to control the operation of the robot, e.g., to
determine oxygen concentration in the fluid around the robot and when to
collect or release oxygen. For the volume used by the pumps, we assume each
pump has size $d=10\,\mbox{nm}$ [29] so the total volume of pumps is $4\pi
r^{2}sd$. We take the controller and other components to require a fixed
minimum volume $v=0.1\,\mbox{$\mu$m}^{3}$ that was estimated for robots
delivering oxygen to tissue [28]. Thus the required volume for components
other than the storage tank is

$v+4\pi r^{2}sd$ (5)

The third constraint is on energy required by the transport robots. Their main
energy use is to fill their tank. This consumes part of the received oxygen,
thereby increasing the filling time. Comparing the pump energy required to
compress oxygen [29] with the rate robots receive oxygen in lung capillaries
(see Section 4.1.1) indicates a robot will only need about $3\%$ of the oxygen
it collects to operate the pumps. Thus filling the tank has a negligible
effect on filling time. Moreover, much of this pumping energy may be recovered
with a generator using the subsequent expansion of the gas when it is released
to the lower partial pressure outside the robot [28].

Robots need energy for their operation after leaving the lung. This includes
sensing when oxygen concentration is low and determining when to release
oxygen, which are tasks that do not depend on robot size. Transport robots
could obtain energy using oxygen from the blood in the same way as the main
robots. However, that would decrease the oxygen available to the main robots,
partially negating the use of transport robots to mitigate oxygen reduction in
the blood. Instead, the transport robots could use their stored oxygen. For
effective oxygen transport, this consumption must be a small fraction of the
stored oxygen so a robot can deliver almost all its oxygen to the main robots.

One measure of this power requirement is the average use over the
$60\,\mbox{s}$ circulation loop. For instance, $0.1\,\mbox{pW}$ is an estimate
of the power needed for computation by robots supplying oxygen to tissue [28].
After transport robots leave the lung, their main activity is determining when
to release stored oxygen, so computation determines their energy demand. Thus
$0.1\,\mbox{pW}$ is a plausible estimate of their required power for
continuous operation. In practice, transport robots would only need to check
oxygen concentrations intermittently, thereby reducing their average power use
below this value. Thus, a reasonable requirement for a transport robot is that
its tank can hold enough oxygen to provide at least $1\,\mbox{pW}$, on
average, during a circulation loop. This provides sufficient energy for the
robot while using only a small portion of its stored oxygen.

Figure 15: Design constraints on oxygen transport robots. The shaded regions
indicate combinations of robot radius and volume fraction for oxygen storage
that violate one or more of the three constraints described in the text. The
black point is the design choice satisfying the constraints with the smallest
value of the total volume of the transport robots given in Eq. 6.

Fig. 15 shows the constraints on robot and tank size for the scenario of
Section 4.1.3, namely when transport robots carry enough oxygen to provide
each main robot with $100\,\mbox{pW}$ for $60\,\mbox{s}$. The figure shows
three constraints. At the upper right, diffusion is too slow to fill the tanks
during a single lung capillary transit. At the left, the robot is too small
and the oxygen tank too large to leave enough room for other robot components.
Finally, at the lower left, the oxygen tank is too small to both provide
enough power to the transport robot during the circulation and deliver most of
its oxygen to the main robots.

Robots satisfying all the constraints, i.e., the unshaded region of Fig. 15,
are feasible designs. Selecting among these designs can optimize operational
or production goals. Operational goals include minimizing the nanocrit, i.e.,
the total volume of the transport robots and reducing their size to simplify
passage through small vessels or slits of the spleen [30]. Production cost
depends on the number of robots and the manufacturing cost of each robot. A
proxy for manufacturing cost is the number of atoms required for their
structure and mechanisms. This proxy is proportional to the volume of the
robot other than the interior of the oxygen tank. Thus a proxy for production
cost is the total volume of all the transport robots multiplied by $1-f$:

$\begin{split}V_{\textnormal{total}}&=n\times\frac{4\pi}{3}r^{3}\\\
V_{\textnormal{production}}&=(1-f)V_{\textnormal{total}}\end{split}$ (6)

property | ratio
---|---
number of robots | $43$
total volume of robots | $1.4$
total manufactured volume of robots | $1.1$
Table 3: Ratio of properties of optimal transport robots to those of the main
robots, The transport robots have radius $r=0.32\,\mbox{$\mu$m}$ and oxygen
volume fraction $f=0.23$, corresponding to the black point in Fig. 15.

An example of optimized parameters are those that minimize the total volume of
the transport robots, $V_{\textnormal{total}}$. These parameters correspond to
the black point in Fig. 15, and also minimize $V_{\textnormal{production}}$ in
Eq. 6. Thus this choice of $r$ and $f$ provides both an operational benefit of
minimizing the nanocrit and a production benefit of minimizing the volume
proxy for manufacturing cost. Table 3 describes the transport robots and how
they compare to the main mission robots.222For comparison, the base design for
gas transport to and from tissue [28] is a robot with $0.5\,\mbox{$\mu$m}$
radius and $50\%$ of the volume used for gas storage, divided equally among
oxygen and carbon dioxide. It devotes about $0.1\,\mbox{$\mu$m}^{3}$ to
components other than the tanks. With these parameters, carrying the same
amount of oxygen per main mission robot as in the scenario of Section 4.1.3
requires about 43 transport robots for each main robot. The main and transport
robots together have a nanocrit about $2.4$ times larger than for the main
robots alone.

|
---|---
(a) | (b)
Figure 16: (a) Cross sections of the main robot (orange, $1\,\mbox{$\mu$m}$
radius) and a transport robot with parameters in Table 3 and oxygen storage
volume indicated by the blue disk. (b) A main robot and nearby transport
robots with blood cells. The main robot, indicated in orange, is near the
center. Transport robots are the smaller white spheres.

Fig. 16 illustrates this optimized design by comparing the oxygen transport
and main mission robots, and a typical placement of robots in a cube of blood
about $20\,\mbox{$\mu$m}$ across. For $10^{12}$ main robots, such a cube
contains, on average, one of these robots (see Table 1), and 43 transport
robots (see Table 3), only a few of which are visible in the figure. The cells
are shown as biconcave shapes [47] with about $10\%$ variation in volume
around a typical red cell volume of $100\,\mbox{$\mu$m}^{3}$ [29]. This
discussion of the number of robots mixed among cells applies to cells and
robots uniformly distributed in a blood vessel whose diameter is larger than
the size of the cube. This is an approximation to the actual distribution
since smaller objects tend to concentrate toward the vessel wall [29].

Due to their larger numbers, the spacing between transport robots in
capillaries is smaller than those for the main robots given in Table 1.
However, the transport robots are also smaller, so their spacing measured in
terms of their size is reduced to a lesser extent. For example, with $10^{12}$
main robots, transport robots are separated by about 8 times their radius in
straight capillaries used for the estimate in Table 1. This suggests that
neighboring transport robots compete to some extent for oxygen in lung
capillaries, which will somewhat increase their filling time compared to the
estimate assuming independent diffusion to each robot. However, lung
capillaries have a complex network structure [84] which could alter the extent
of this competition, especially due to the variation in paths and transit
times. Extending studies of how cells move through these vessels [77] to hard
particles of size comparable to these robots could quantify this effect.

For the transport robots shown in Fig. 16, the pump volume is a small fraction
of the component volume in Eq. 5. If pump size and surface coverage are larger
than assumed here, an additional optimization is allowing for a smaller number
of pumps than indicated by Fig. 14. This trades a longer filling time due to
fewer pumps for additional volume for other robot components. This would give
another parameter to optimize, in addition to robot radius and tank size shown
in Fig. 15.

The specific optimal robot and tank sizes shown in Fig. 16 depend on the robot
component size and power requirements (i.e., the second and third constraints
in Fig. 15). This example illustrates how multiple design constraints combine
to determine the choice of robot parameters. These constraints arise both from
the robot’s external environment (e.g., the diffusion rate of oxygen) and the
robot’s internal capabilities (e.g., the power required for its operation).
Better estimates of these parameters require more detailed design of the robot
components. Depending on these values, there could be no feasible design,
i.e., no unshaded region in Fig. 15. That would indicate the robots’ component
volumes or power requirements are too large to provide this oxygen. In that
case, oxygen transport robots could provide less than the $100\,\mbox{pW}$
used in this scenario, or the main robots could use another mitigation
strategy to avoid low oxygen concentration.

Using transport robots increases nanocrit. While likely not a significant
issue for the scenarios of Table 1, large nanocrit from transport robots could
alter blood flow [29] or require a compensatory reduction in the number of
main mission robots. Moreover, increasing the number of robots collecting
oxygen in the lungs will eventually extract all available oxygen. At this
point, additional robots will not increase the total collected oxygen and
would reduce the reserve available to the body by increasing blood perfusion
in the lungs.

Transport robots carry oxygen from the lungs to the systemic circulation. In
the case of passive flow, transport robots release oxygen into the surrounding
blood plasma when they detect low oxygen concentrations. Some of that released
oxygen diffuses to nearby robots. The rest remains in the blood plasma or
diffuses into tissue or red cells. This means the blood carries some of the
released oxygen back to the lung without providing robot power. This wasted
oxygen is particularly significant when released in the veins, where it will
not diffuse into tissue. This is a likely scenario since the lowest
concentrations occur in veins (see Fig. 6a). This diffusive oxygen delivery by
specialized robots is not as effective as when robots carry their own oxygen,
as described in Section 4.1.3.

Robots could partially offset the limitation of diffusive transport by
releasing oxygen only when they are close to a main mission robot, as
determined, for example, via short range communication [29, 41]. This
proximity allows more of the released oxygen to reach the receiving robot’s
surface. If necessary to support high burst power, multiple transport robots
could aggregate around a main one and simultaneously release oxygen. This will
temporarily produce a high oxygen concentration in that region. However, as
not all of that oxygen will reach the robot, the total release must be limited
to avoid tissue damage due to excessive oxygen, i.e., hyperoxia, which is a
particular concern in the brain [87]. This safety limit due to released oxygen
not reaching a main robot limits how rapidly a group of transport robots can
deliver oxygen, thereby reducing their ability to support high burst power
that robots could obtain if they carry their own oxygen.

For greater efficiency, oxygen transport robots could directly transfer oxygen
to the main robots by docking with them. Direct transfer provides oxygen in
the same way as an onboard tank. In this approach, transport robots act as
external oxygen tanks for the main robots. Using separate robots to carry
oxygen allows them to selectively provide power to robots that most need it.
For example, this selectivity could allow one robot in a group to have a burst
of high power for long range communication of a summary of data collected by
the group. While this flexibility is a potential benefit, some amount of power
variation within a group of robots can also be achieved if most robots limit
their oxygen demands, as described in Section 4.2.

Direct transfer requires more complicated robots than releasing oxygen into
the blood and relying on diffusion. In particular, docking requires locomotion
and navigation on the part of the transport or main robots, or both, to find
each other in the constantly changing fluid environment as the robots and
cells move. If the main robots need locomotion capability for their mission,
they could also use that capability to reach oxygen transport robots. In that
case, there is no need for the transport robots to also have locomotion
capability, thereby simplifying their design and providing more room for them
to carry oxygen. An additional issue is if a transport robot delivers oxygen
by completely emptying its tank while docked, direct oxygen delivery would
produce a population of transport robots with a declining fraction of those
with full tanks. So robots with full tanks will become harder to find and may
need a communication protocol to identify transport robots that have available
oxygen.

####  4.1.5 oxygen reservoirs formed by specialized robots

Fig. 5 shows $10^{12}$ robots deplete oxygen toward the end of the circulation
loop, i.e., in veins. One approach to mitigating this oxygen depletion is to
deliver oxygen reservoirs to those vessels. These reservoirs could be filled
with oxygen prior to their placement in the body, and thereby supplement
oxygen available via the lungs until the oxygen they carry is consumed.
Examples of such single-use devices include bubbles enclosed in biocompatible
particles of sizes from about $100\,\mbox{nm}$ to several microns [45] that
have been used to enhance ultrasound imaging in the body. These particles can
release oxygen gradually by diffusion or rapidly in specific areas by
disrupting them with resonant ultrasound. In addition, implanted biodegradable
microscopic tanks that release oxygen by diffusion have been experimentally
demonstrated in tissue scaffolds [18]. By relying on diffusion, these tanks
require no power or control, but their rate of oxygen release decreases
exponentially with time. The demonstrated time constants for this decrease is
several hours, which could support robot missions of such durations, but the
decreasing oxygen release could limit how well oxygen delivery from a
reservoir matches robot demand. The development of more sophisticated oxygen
delivery robots [28] could provide a reservoir that supplements oxygen from
the lungs in a more effective way.

A rechargeable reservoir with control over oxygen release could be formed from
robots small enough to travel through the circulation and able to attach to
vein walls. These would be the same type of circulating oxygen-carrier robots
discussed in Section 4.1.4 but used as a reservoir fixed to vessel walls
rather than traveling in the blood along with the main mission robots.

As an example of this scenario, consider a reservoir capable of supplying
enough oxygen for $100\,\mbox{pW}$ to $10^{12}$ robots during the last
$20\,\mbox{s}$ of their circulation. The robots in the reservoir could do so
by releasing oxygen into the blood, but only a portion of that would diffuse
to the robots, the rest would return to the lung unused. Instead, suppose
circulating robots can find and dock with the reservoir robots to receive
oxygen directly.

Due to the uniform distribution of the main mission robots considered here,
the last $20\,\mbox{s}$ of the circulation contains about a third of these
robots. To provide these robots with sufficient oxygen to generate
$100\,\mbox{pW}$, the reservoir would need to supply
$10^{20}\,\mbox{molecule}/\mbox{s}$. To maintain this rate for a single
circulation time, the reservoir would need to start with $6\times 10^{21}$
oxygen molecules. As an example, suppose the reservoir consists of
$1\,\mbox{$\mu$m}$-radius robots using $80\%$ of their volume to store oxygen
in tanks with $20\,\mbox{nm}$-thick walls and at one-third the tanks’ maximum
pressure. Such a robot could store $4.6\times 10^{10}$ molecules and the
exterior volume of the tank, i.e., including the shell, would occupy $85\%$ of
the robot’s volume. If a reservoir robot uses $10\,\mbox{pW}$ to support its
operation, it would consume about $4\%$ of its stored oxygen during a
$60\,\mbox{s}$ operation time. Thus the reservoir would require $1.4\times
10^{11}$ robots to provide oxygen to the $10^{12}$ main mission robots.

A reservoir robot with these parameters would require $26\,\mbox{s}$ to fill
its tank in a lung capillary, compared to the typical $0.75\,\mbox{s}$ transit
time through a lung capillary. If the robots move passively with the
circulation rather than concentrating in lung capillaries for the required
time, and the robots do not use power while circulating, each would require
$35$ circulations through the lung to fill its tank, which would take about
half an hour.

Thus in this scenario, using a reservoir would provide oxygen to support
$100\,\mbox{pW}$ for robots during the last third of a circulation loop only
once in every $35$ circulations. In the remaining circulations, the robots
would have little power (see Fig. 5) during the last third of their
circulation, or no power if they stop using power to avoid extremely low
oxygen concentration and cell saturation toward the end of the circulation
loop (see Fig. 6).

####  4.1.6 red blood cells as oxygen reservoirs

Red blood cells are an oxygen reservoir in the blood. Robots, which are in the
blood plasma, extract oxygen from nearby red cells indirectly: robots consume
oxygen in the plasma, lowering the concentration in the plasma, which leads to
partial replenishment with oxygen released from hemoglobin in the cells. Only
some of the oxygen released from cells diffuses to nearby robots.

The robots considered here have volume of a few cubic microns, which is
considerably smaller than the typical $100\,\mbox{$\mu$m}^{3}$ volume of a red
blood cell. Thus, if robots are able to enter red blood cells, robots could
extract oxygen directly from hemoglobin, e.g., by collecting oxygen-bound
hemoglobin in a tank from which oxygen is continually pumped out. This process
would be limited by the kinetics of oxygen release from hemoglobin [16] rather
than the diffusion rate of oxygen in plasma. The robot could promote this
oxygen release by altering the chemical environment of the hemoglobin in its
tank, e.g., by adding carbon dioxide.

This use of red blood cells as reservoirs could increase the rate robots
obtain oxygen compared to operating in plasma. However, in cases where robots
in plasma consume most of the oxygen by the end of the circulation loop, this
would not provide more oxygen in total during the circulation. Instead, it
would shorten the time until robots deplete the oxygen. Moreover, if robots
inside cells reduce cell saturation below the equilibrium value with the
surrounding plasma, then cells will absorb rather than release oxygen into the
plasma. When passing through capillaries, this would lead to cells competing
with tissue for oxygen, rather than replenishing oxygen in the plasma in
response to tissue consumption. Evaluating the extent to which this occurs
could be done by extending the model to include direct reduction in hemoglobin
saturation within cells based on the robot consumption rate.

###  4.2 Limiting Robot Power

Fig. 5 shows a large variation in the power available to robots as they move
through the circulation. Robots could mitigate their effect on oxygen
concentration by limiting their power use, particularly early in the
circulation loop, i.e., in arteries, where oxygen is plentiful. This section
discusses some strategies robots could use for limiting power. Since $10^{12}$
robots consuming oxygen as fast as possible leads to significant oxygen
reduction, we focus on this case for evaluating the consequences of limiting
power use.

A fixed limit on robot oxygen consumption could be implemented in hardware by
reducing the number or capacity of pumps on the robot surface, or fuel cells
inside the robot. This would prevent the robot from using all oxygen reaching
its surface when the concentration is high, i.e., shortly after leaving the
lungs. The more flexible and targeted power limitation methods discussed in
this section require robots to alter pump capacity based on sensor
measurements of quantities such as oxygen concentration, location in the body
or distance to vessel walls. Moreover, software-based limits would allow some
robots to use all the oxygen when occasions arise that could benefit from high
burst power.

An alternative method for limiting oxygen collection in large vessels is for
the robots to travel in groups, analogous to the grouping of blood cells in
some blood vessels [29]. Such groups could form from robots waiting in
moderate-sized vessels, such as lung veins, for a group to accumulate, or
robots could search for others to form a group while moving with the blood. If
the group is large and compact enough that some robots are completely
surrounded by others, the robots at the surface of the group could share
collected oxygen with those inside, or the robots could occasionally change
positions so each robot spends part of the time at the surface of the group.
When vessels become too small for the group, the robots could disperse into
smaller groups or individual robots, adjusting the size to match that of the
vessels [24].

Grouping reduces absorption per robot by exploiting the competition for oxygen
among nearby robots, which is the opposite of the improved absorption by
smaller robots discussed in Section 4.1.4. For example, suppose $n$ robots
join to form a spherical group of radius $R$. To accommodate the volume of
these robots, $R\approx r_{\textnormal{robot}}\sqrt[3]{n}$. From Eq. 1, the
rate this group collects oxygen is proportional to $R$. Thus the oxygen
collected per robot in the group is reduced from that of isolated robots by a
factor of $(R/r_{\textnormal{robot}})/n\approx n^{-2/3}$. For example, 100
$1\,\mbox{$\mu$m}$-radius robots could form a group with $R\approx
5\,\mbox{$\mu$m}$ and then each robot would receive about $5\%$ of the oxygen
that isolated robots collect. For this method of power reduction, the groups
need not all have the same size nor contain all the robots in large vessels.
For example, if among 200 robots, 100 form a group while the others remain
isolated, the total oxygen consumed by those robots would be about half what
they would consume without any grouping.

Figure 17: Power for $10^{12}$ robots during a circulation loop. The figure
compares three cases: 1) unlimited power (the same as shown in Fig. 5), 2)
robots limited to a maximum of $200\,\mbox{pW}$, and 3) robots limited to
$20\,\mbox{pW}$ in arteries, $200\,\mbox{pW}$ while in capillaries, and
unlimited use in veins. For the two cases with limited power, the dashed
curves show the power available to a robot if it consumed all oxygen reaching
its surface (provided few, if any, other robots exceed their limits so oxygen
is not significantly reduced by the robot switching to maximum power). The
light vertical lines indicate when the blood is in a capillary. Figure 18:
Oxygen concentration for $10^{12}$ robots. The figure compares the same three
cases as shown in Fig. 17. The light vertical lines indicate when the blood is
in a capillary.

####  4.2.1 Power Limits for All Robots

The simplest approach to limiting robot oxygen consumption is for all of them
to have a fixed maximum power generation rate. Fig. 17 shows one example of
this approach for $10^{12}$ robots where a $200\,\mbox{pW}$ limit extends the
range of a circuit where robots have power, but power still falls to zero
before returning to the lung. To provide some power throughout the circulation
loop with this many robots would require a somewhat lower power limit.

The dashed curves in Fig. 17 indicate power available to a robot from the
oxygen concentration in the plasma around that robot. This is the power a
single robot would have if it switches from limited-power to using all
available oxygen. If only a few robots make this choice, they will not
significantly lower the oxygen. Thus the dashed curve indicates the power
available to a few robots, e.g., for occasional burst activity, provided at
most a tiny fraction of the robots make use of that additional power at any
given time. An example of such activity is the initial safety evaluation
discussed in Section 3.4. Any robots detecting problems could use the high
available power for burst communication of its measurements.

By limiting their power production when oxygen is plentiful, robots leave more
oxygen for robots later in the circulation loop. In effect, the robots use
blood cells as external oxygen storage tanks to shift when and where robots
utilize oxygen. This is conceptually similar to oxygen provided by additional
transport robots discussed in Section 4.1 without the need for additional
robots. On the other hand, specialized transport robots could deliver oxygen
more effectively, especially if they use pumps for transfer rather than
relying on diffusion.

Instead of a fixed power limit, robots could use a specified percentage of the
oxygen reaching their surface. Unlike a fixed power limit, such as
$200\,\mbox{pW}$, a percentage limit would adjust to the decreasing oxygen
concentration as robots move through a circulation loop. For example,
$10^{12}$ robots using only $10\%$ of the oxygen would have the same effect on
oxygen concentration as $10^{11}$ robots using all available power, as
illustrated in Fig. 6. The power available to each robot would be one-tenth
that shown for $10^{11}$ robots in Fig. 5, i.e., the robots would have a few
tens of picowatts. This would only be a worthwhile alternative to using
$10^{11}$ robots if there is a benefit of having ten times as many robots,
each with one-tenth the power. For instance, if the mission is to have at
least one robot pass through every capillary to look for a target location,
e.g., recognized by a rare pattern of chemicals, and this detection can be
done with low power. In that case, a larger number of robots will complete the
survey more rapidly than a smaller number of robots. Moreover, if the response
to the detection requires much more power, that will be available to the few
robots finding the target due to the higher oxygen concentration left by most
robots using only a small fraction of the available power.

The above discussion of power limits supposes those limits are always in
effect. Another possibility is limits that apply occasionally so robots
alternate between consuming much or all of the available oxygen and limiting
their consumption. For example, robots could consume significant oxygen only
every second or third circuit. If such duty cycling occurs independently among
robots, it would reduce the number of active robots by the corresponding
factor, i.e., 2 or 3 in this example. On the other hand, if robots synchronize
their schedules, oxygen in the circulation would alternate between high and
low levels. This could be beneficial if the harm from continuous moderately
low oxygen is worse than switching between very low and normal concentrations.

####  4.2.2 Power Limits Based on Location within a Circulation Loop

A more sophisticated limitation method is for the limits to depend on the
robot’s location rather than using a single overall limit. Such a strategy
would be particularly useful if the main operation of a robot occurs when it
is in a capillary, e.g., measuring properties of nearby tissues. In this case,
robots could reduce power use while in arteries, to have more available for
the short time they spend in capillaries. They could also defer some power
consumption, e.g., analyzing measurements they collected in the capillary,
until they reach vessels with more available oxygen. This could be in veins,
where additional reduction in oxygen saturation of red cells would no longer
affect oxygen available to tissues. This strategy would require some
adjustment for portal flows, e.g., in the portal vein, where blood flows
through another capillary, in the liver, before returning to the lungs. In
that case, the robot could wait until it has passed a liver capillary before
increasing its power use.

Fig. 17 shows an example for $10^{12}$ robots using this method. With the
parameters illustrated here, this approach provides power throughout the
circulation loop. In particular, it provides higher power when robots are in
the veins compared to robots either using all available power or limiting
their use to at most $200\,\mbox{pW}$. Fig. 18 shows the effect of this limit
on oxygen concentration.

Implementing location-dependent limits on power requires that robots determine
the type of vessel they are in. Robots could do so in a variety of ways, with
trade-offs between complexity for robot processing and accuracy. One approach
is to use an onboard clock to measure the time since a robot left the high-
oxygen environment of a lung capillary. They could use a fixed time, e.g., 30
seconds, to decide when they have likely reached a vein and can increase power
use. This approach will not adjust for variation in circulation speed due to
changes in heart rate, nor variations in circulation path lengths, but is a
simple approach that avoids the need for robots to determine when they have
passed through a capillary.

Alternatively, robots could measure oxygen concentration in the surrounding
plasma. This is high in arteries, decreases as the robot passes through a
capillary where tissue consumes oxygen, and is relatively low in veins.
Concentration thresholds required to distinguish these types of vessels
depends on the concentration of robots (see Fig. 6a) and the power-consumption
method of the robots (see Fig. 18). This variation could be predefined based
on these choices. More challenging for estimating location from oxygen
concentration is the variation in tissue use in different organs and at
different times, and variation in hematocrit in small vessels. These
variations limit the accuracy of using oxygen concentration to determine the
type of vessel the robot is in.

Combining a variety of measures can identify vessel type more accurately. Most
circuits through the body move through arteries of decreasing size, through a
capillary, and then through veins of increasing size. Changes in pressure and
how much it varies over the duration of a heartbeat distinguishes arteries
from veins [29]. For small vessels, changes in fluid flow near the robot allow
estimating vessel size [38].

####  4.2.3 Power Limits Based on Location in the Body

In addition to adjusting power based on the type of vessel a robot is in, it
could adjust its power based on its macroscopic location in the body. For
example, robots could use information on which organ they are in [29] to set
their power limits. This would allow robots to adjust power generation to
match organ-specific tasks. As an extreme case, if robots only need to be
active in one organ, then a number of robots large enough to deplete oxygen if
they all use power would instead consume much less oxygen and mainly affect
that organ and tissue downstream from that organ. This would avoid depleting
oxygen in veins throughout the body, but could lead to significant local
oxygen reductions, particularly in the small veins leaving those organs before
the blood reaches larger veins, where it mixes with blood from other organs.

As seen in Fig. 6, the most severe reduction in oxygen with $10^{12}$ robots
occurs near the end of the circuit, i.e., after blood has mixed into large
veins. For robots limiting power use to one or a few organs, that extreme
reduction would not occur: instead, blood from other organs would partially
restore oxygen in the vein. Thus organ-specific power use could tolerate
larger numbers of robots than if all robots consume oxygen. A trade-off in
this scenario is that only the portion of robots in the target organ at any
given time would be actively performing their tasks while the majority of
robots simply move passively with the blood. Alternatively, robots with
locomotion could target the organs of interest, thereby providing sufficient
concentration to perform their mission with a smaller total number of robots.

Another application for power limits based on location within the body is to
adjust to organ-specific variations in tissue demand, beyond those compensated
for by variation in tissue capillary density or changes in blood flow in
response to those variations. That is, robots may need to reduce their oxygen
consumption in tissue with higher than usual oxygen demand. At a local scale,
robots could determine such limits from measuring oxygen concentration.
However, by the time a robot reaches the tissue, and encounters the lowered
oxygen concentration, it will be too late for limiting power in small arteries
leading to that tissue and where oxygen concentration may still be relatively
high. In this case, a robot could limit power more effectively using
information on which organ it is entering and the overall tissue demand of
that organ.

####  4.2.4 Power Limits Based on Robot History

Instead of a power limit on all robots, the limit could be selective by
depending on the recent history of each robot. The benefit of a history-
dependent limit depends on the fraction of robots requiring significant power
at any given time, as determined by their current configuration and their
local environment. For example, if only 10% of robots need significant power,
then the effective number consuming oxygen is reduced by that fraction. If
this reduction is, at least roughly, uniform throughout the body, the robot
power will affect oxygen according to the effective number of active robots.
For instance, if only 10% of a trillion robots generate significant power and
do so at their maximum possible rate, the effect on oxygen concentration will
correspond to $10^{11}$ robots consuming oxygen as rapidly as they can.

One situation leading to history-dependence arises in robots with oxygen
storage and considerable variation in power requirements, depending on where
they travel in the body. Such a robot could limit its oxygen intake and
supplement that intake with oxygen from its storage tank. When the tank is
nearly empty, the robot could absorb more rapidly to fill it. This would be
useful if neighboring robots share data and only one of them needs to expend
significant power to process or communicate the data: the robot using burst
power would deplete its tank and need to refill it.

Another case of history-dependent power is if robots only need high power when
handling rare events. This could occur when a robot detects a specific
chemical in the blood and needs to move toward its source [37]. Or if such a
robot requires confirmation from other nearby robots before taking action, to
reduce the number of false positives. In this situation, the detecting robot
could increase its power use to communicate with nearby robots and evaluate
their responses. In this case, the originating robot may need sufficient power
to send a significant number of bits, e.g., a summary of the data it
collected, whereas responding robots could just send a short response
indicating whether their information is consistent with that from the
detecting robot.

Figure 19: Markov model of amount of data stored by a robot with capacity to
store data from $5$ capillaries and ability to transmit the data from up to
$3$ capillaries each time it reaches the skin. The number in each node is the
number of capillary measurements the robot has currently stored. The thickness
of the edges correspond to the transition probabilities. For this
illustration, robots do not collect data from the lung or the skin.

As a quantitative example of history-dependent power, consider robots that
measure chemical concentrations in capillaries and communicate their readings
when they are in range of external receivers on the skin. For example,
receivers could emit a beacon signal that robots can detect when they are
nearby. Suppose the relevant data for this mission occurs in tissue other than
the lung or skin. In this case, the relevant property of each circulation loop
is whether the robot goes to the skin, and, if not, whether it flows through a
portal system, thereby measuring two capillaries before returning to the lung
for another loop through the circulation. With typical resting perfusion rates
[25], $8\%$ of the blood flows to the skin and $20\%$ flows through the portal
vein, which is the major circulation path passing through two capillaries. The
rest of the blood flows through a single capillary that is not in the skin
before returning to the heart. Suppose a robot can store measurements from up
to $5$ capillaries, and has time to transmit the data from up to $3$
measurements while near the skin.

Due to the mixing of blood in the heart, where a robot goes in the body during
each circulation loop is independent of where it went previously. This leads
to a Markov stochastic process for the amount of data stored by a robot with
the transition graph shown in Fig. 19. Each edge in the graph corresponds to
the robot making a single circulation through the body. For instance, a robot
with empty data storage is most likely to store data from one capillary during
its next circulation. A robot could also go to the skin, in which case it does
not collect data, or through a portal system, in which case it collects data
from two capillaries.

After multiple loops, the Markov process approaches its stationary
distribution, in which about $70\%$ of the robots have filled data storage.
Such robots are unable to collect any additional data until they have had an
opportunity to transmit, and delete, some of their collected data during their
next passage near the skin. Moreover, most of the robots reaching the skin
would have data on at least 3 capillaries, and hence have enough data to
transmit at their maximum rate while near the skin.

Suppose the robots mainly require power during data collection. The $70\%$ of
robots with full data storage then use only a minimal amount of power. In
terms of power use, $10^{12}$ robots would be comparable to $3\times 10^{11}$
robots consuming oxygen as fast as they can. This would be similar to an
overall limit of robots using $30\%$ of available power, but with history
dependence so the power use is targeted to those robots that most need it,
rather than a reduction for all robots.

####  4.2.5 Limiting Oxygen Consumption Near Vessel Walls

Figure 20: Oxygen concentration along the wall of a straight vessel of
$2\,\mbox{mm}$ diameter with $10^{12}$ robots employing different oxygen
consumption methods: full consumption of all oxygen reaching the robot
surface, no consumption by robots within $0.3\,\mbox{mm}$ of the vessel wall,
and all robots limiting their consumption to $50\%$ of available oxygen.

The above location-based limits considered adjusting power consumption based
on a robot’s position in the circuit, corresponding to the time since the
robot was last in the lung, and based on the type of tissue or organ the robot
is in. These approaches can increase the amount of oxygen remaining in the
blood, particularly toward the end of the circulation.

As described in Section 3.1, low oxygen concentrations in veins could affect
cells on the vessel wall. Addressing this issue suggests another location-
based limit, namely for robot power limits to apply only when a robot is close
to the vessel wall. This could be useful in vessels whose diameter is at least
a millimeter or so since, for vessels of that size, diffusion limits oxygen
transport to a relatively small fraction of the vessel diameter during the
time a robot passes through the vessel. Specifically, Fig. 6 shows the low
oxygen concentration with $10^{12}$ robots occurs during the last 10 to 20
seconds of the circulation loop. The characteristic diffusion distance of
oxygen during $t=20\,\mbox{s}$ is
$\sqrt{2D_{\textnormal{$\mbox{O}_{2}$}}t}=0.3\,\mbox{mm}$, with
$D_{\textnormal{$\mbox{O}_{2}$}}$ given in Table D.1. Thus robots could avoid
creating low concentration near vein walls by reducing their oxygen
consumption when they are near the vessel wall, e.g., as determined by
measuring fluid stresses on their surfaces [38]. Alternatively, if robots have
locomotion capability, they could move away from the vessel wall. In this
case, the low oxygen concentration seen in Fig. 6 would occur in the central
portion of the vessel only.

The effectiveness of limiting oxygen consumption based on distance to the
vessel wall depends on how much mixing occurs during transport in moderate-
sized veins, including the effect of merging vessels and cell motion. For
veins whose diameter is substantially larger than the size of cells, this
could be evaluated by approximating the blood as a uniform fluid. For smaller
vessels, this evaluation requires simulations including deformable cells,
e.g., in vessels with diameters up to hundred microns or so [4]. The main
advantage of limiting power use near vessel walls arises in vessels large
enough that oxygen does not have time to diffuse across the vessel, and there
is not significant mixing from the flow. In general, mixing is slow in laminar
flow [76] found in such vessels.

As a quantitative example of this mitigation method, consider the flow in a
$40\,\mbox{mm}$-long portion of a vein with diameter of $2\,\mbox{mm}$.
Suppose oxygen is well-mixed through the blood as it enters this vessel
segment, has a relatively low concentration of $0.5\times
10^{22}/\mbox{m}^{3}$ and there are $10^{12}$ robots in the circulation. This
example corresponds to the situation at about $45\,\mbox{s}$ in Fig. 6, and
oxygen is fully depleted after a few more seconds of full consumption by the
robots.

The model developed here for robot power over a full circulation loop treats
the flow as one-dimensional and gives the average concentration over the
vessel cross section. For a more detailed view of oxygen concentration, this
example evaluates oxygen transport in the vessel segment with parabolic, i.e.,
Poiseuille, flow profile and with average speed of $2.5\,\mbox{mm}/\mbox{s}$.
Oxygen transport is a combination of convection with that flow, diffusion,
consumption by robots and replenishment by red cells, as described in Appendix
C. This vessel is a vein, so there is no consumption by tissue. In this case,
the mixing due to the motion of blood cells is a minor addition to the
diffusion of small molecules, such as oxygen, in the blood [29]. From Eq. B.1
the Peclet number for this oxygen transport is $\mbox{Pe}=2500$, so oxygen
mainly flows along the vessel with relatively little diffusion across it. This
behavior leads to considerable variation in oxygen concentration across the
vessel: robots in the slowly moving blood near the walls deplete oxygen much
more than robots traveling with the faster flow near the center of the vessel.

Robots could mitigate the rapid concentration decrease near the wall by
reducing their oxygen consumption. As an example, Fig. 20 shows how the
concentration near the vessel wall changes in three cases. In the first case,
robots fully consume oxygen reaching their surface. In the second case, robots
within $0.3\,\mbox{mm}$ of the vessel wall do not consume oxygen, with that
distance corresponding to oxygen’s characteristic diffusion distance discussed
above. This distance to the vessel wall accounts for about half the total
cross sectional area of the vessel. Thus in this case, only about half the
robots are consuming oxygen, i.e., those in the central portion of the vessel.
This leads to the third case shown in Fig. 20 where only half the robots
consume oxygen, or, equivalently, each robot consumes only half the oxygen
reaching its surface, without regard to their position in the vessel. As seen
in Fig. 20, this overall reduction in power reduces the rate of oxygen
depletion near the wall, but is much less effective than when robots limit
power based on their distance to the wall.

Figure 21: Example of merging vessels: the branches, each with diameter of
$2\,\mbox{mm}$, merge into a vessel with diameter of $2.5\,\mbox{mm}$. Numbers
along the axes are positions in millimeters measured from an origin at the
center of the merging vessels. Figure 22: Oxygen concentration on a cross
section through the merging vessels on the symmetry plane with $10^{12}$
robots in the bloodstream, each consuming all oxygen reaching its surface.
Figure 23: Oxygen concentration along the upper wall of the merging vessels
with $10^{12}$ robots employing different oxygen consumption methods: full
consumption of all oxygen reaching the robot surface, no consumption by robots
within $0.3\,\mbox{mm}$ of the vessel wall, and reduced consumption
corresponding to the fraction of robots at least $0.3\,\mbox{mm}$ from the
vessel wall, which is about 50% in these vessels. Figure 24: Oxygen
concentration on a cross section through the merging vessels on the symmetry
plane with $10^{12}$ robots in the bloodstream. Left: no consumption by robots
within $0.3\,\mbox{mm}$ of the vessel wall. Right: robots consume 50% of
oxygen reaching their surfaces.

The example of Fig. 20 is for flow in a single straight vessel. The flow from
the merging of veins of various sizes could somewhat increase the mixing and
thereby reduce the benefit provided by robots limiting consumption near the
wall. To evaluate this effect, Fig. 21 is an example of merging vessels with
asymmetric branching. As is typical of merging blood vessels, the total cross
section of the two branches exceeds that of the main vessel, so flow speed in
the branches is somewhat slower than in the main vessel. To focus on the
effect of the merging vessels, the example includes only $8\,\mbox{mm}$ in
each branch and the same distance in the main vessel. This is considerably
shorter than the $40\,\mbox{mm}$-long straight vessel discussed with Fig. 20.

Accounting for the effect of the branches requires solving for the fluid flow
through the vessels. As with the previous example, we suppose parabolic flow
exits the main vessel with average speed $2.5\,\mbox{mm}/\mbox{s}$ and the
inlet pressure at both branches is the same. The fluid flow with these
boundary conditions is laminar. Fig. 22 shows the resulting concentration
distribution on a cross section through the vessels along their symmetry
plane. The merging flows provides some mixing where the branches join.
Nevertheless, the behavior of the concentration is similar to that seen in the
straight-vessel example. Specifically, Fig. 23 shows the oxygen concentration
along the upper wall of the upper branch and main vessels for the same three
cases as shown for the straight vessel in Fig. 20. Fig. 24 compares the
concentration across the vessels for the two cases where oxygen consumption is
limited.

Even when robots near the vessel wall do not consume oxygen, the concentration
near the wall eventually gets very small. Thus, in the context of these
examples, the main benefit of this mitigation is extending the range of the
circulation loop by a few centimeters compared to when robots consume all
oxygen or limit consumption to $50\%$. This increase in range could be
especially beneficial in a vein just before it merges with other veins that
contain blood from shorter circuits and hence have more remaining oxygen than
the model estimates for an average circulation loop in Fig. 6. In that case,
avoiding fully depleted oxygen for a few additional centimeters could be
sufficient to avoid extremely low concentrations near the walls of any veins,
without requiring all robots in those vessels to reduce their power
generation. Alternatively, achieving the same increase in range with a power
limit on all robots would require a much larger reduction in power than the
$50\%$ reduction that corresponds to robots near the wall consuming no oxygen
while the rest of the robots consume at their maximum rate.

The one-dimensional flow model used here assumes oxygen and cells are well
mixed over the vessel cross section over the distance and time scales of
interest here. In small vessels, diffusion is fast enough to provide this
mixing. In larger vessels, mixing depends on the nature of the flow, including
enhanced mixing due to motion of the cells moving in the blood, changing
vessel diameters and mixing flows from branches. Those fluid behaviors are not
sufficient to fully mix oxygen in millimeter-sized vessels. Thus another
consequence of limiting consumption near walls is increasing the uniformity of
the concentration across the vessel. In this case, the robot behavior extends
the accuracy of the model’s assumption of uniform mixing to millimeter length
scales. This is an example of robots adjusting their behavior to improve the
accuracy of a simple global model of their behavior. This simplification
arising from robot behavior can be useful when applying such models to
evaluate global behaviors and select among different control protocols for the
robots.

###  4.3 Positioning Robots in Capillaries

The small volume of capillaries leads to relatively large variation in robot
numbers as described in Section 3.3. Those capillaries that contain
significantly more robots than average will have less power. Robots could
reduce this variation by moving away from particularly close neighbors,
including moving to other capillaries that have fewer robots.

Within a single vessel, robots in close proximity compete for oxygen. Such
robots could improve oxygen distribution using a small-scale version of the
power limiting strategies discussed in Section 4.2. In addition, the flow of
oxygen to robots depends on their positions along the vessel wall, as shown by
the streamlines in Fig. 11. This position dependence provides an additional
mitigating strategy for small groups of nearby robots: improving power
distribution by deliberately adjusting their positions relative to their
neighbors.

Fig. 11 suggests there are competing effects on the oxygen delivered to
robots. On the one hand, a robot directly upstream of another absorbs much of
the oxygen that would otherwise go to the downstream robot. However, placing
all robots on one side of the vessel allows more oxygen to reach downstream on
the other side of the vessel, which can then diffuse across the vessel to
downstream robots. The relative importance of these effects depends on the
ratio of convective to diffusive transport of the oxygen, i.e., the Peclet
number of the oxygen transport [76].

Figure 25: Example positioning of 5 robots in a vessel with $8\,\mbox{$\mu$m}$
diameter viewed from the vessel inlet. Successive robots are offset around the
wall by the angle $\theta$ and spaced along the vessel by
$5.5\,\mbox{$\mu$m}$. The arrangements shown in the top and bottom of Fig. 10
correspond to $\theta=180^{\circ}$ and $0^{\circ}$, respectively. Figure 26:
Relative power for 5 robots in a vessel as a function of angle $\theta$ with
the geometry of Fig. 25. Each curve shows the power relative to the situation
with all robots aligned along the vessel wall, i.e., with $\theta=0$. The
solid curves show the average power for the 5 robots with average fluid speed
$1\,\mbox{mm}/s$ and $0.2\,\mbox{mm}/s$ for the upper and lower curves,
respectively. The dashed curves show the power available to the fifth robot,
i.e., the robot farthest downstream in this group, for these fluid speeds.

To illustrate the potential of position adjustment, consider 5 robots
positioned along a vessel wall with neighbors offset by an angle $\theta$, as
shown in Fig. 25. This arrangement is analogous to the angular spacing of
leaves around the stem of a plant, where angles related to the golden ratio
can minimize the extent to which higher leaves cast shadows on lower ones
[78]. However, unlike the direct path of light, diffusion allows some oxygen
to reach robots that are directly downstream of others on the vessel wall.

Fig. 26 shows the offset angle has different effects on the average power for
these robots than it does for the last robot, which receives the least oxygen.
Specifically, offsetting neighboring robots by $180^{\circ}$ increases the
average power by a few percent, with larger effect for faster moving fluid.
This corresponds to the top arrangement in Fig. 10. On the other hand,
positioning all robots on the same side of the vessel (i.e., $\theta=0$)
increases power for the last robot even though it is directly downstream of
all the other robots in the group. This occurs because robots on one side of
the vessel allow more oxygen to reach downstream along the other side of the
vessel and then diffuse to the last robot. As seen with the streamlines in
Fig. 11, this effect is larger for slower moving fluid. The robots could
select among these options based on the relative importance of maximizing
power for the group as a whole compared to ensuring all robots have at least a
minimum amount of power, and depending on the size of the vessel and the speed
of the fluid.

Figure 27: Relative oxygen concentration for nonuniformly-spaced robots on a
vertical slice through the center of the vessel and the robots. The white
curves are streamlines of the oxygen flux. The rows correspond to the robot
positions shown in Fig. 10. The fluid flows from left to right in each
diagram, with average speed $0.2\,\mbox{mm}/\mbox{s}$ and
$1\,\mbox{mm}/\mbox{s}$ in the left and right columns, respectively.

Robots could use nonuniform positions along the vessel to provide more power
to downstream robots by having upstream robots closer to each other than
downstream ones. As an example, in addition to the angular separation around
the vessel discussed above, suppose the robot positions along the direction of
the vessel increase quadratically, starting with a difference of
$2.5r_{\textnormal{robot}}$ for the first two robots and occupying the same
total distance along the vessel as the uniformly spaced robots discussed
above. In the case of zero offset angle between neighbors (i.e., $\theta=0$),
the distance between facing surfaces of the first two robots is
$0.5r_{\textnormal{robot}}$. Fig. 27 shows the oxygen concentration for
nonuniformly spaced robots when successive robots are on opposite sides and
the same side of the vessel. The larger distance between the last robot and
the others allows the last robot to collect oxygen from a larger portion of
the fluid than when robots are uniformly spaced (Fig. 11).

Figure 28: Relative power for 5 robots in a vessel as a function of angle
$\theta$ with nonuniform spacing along the vessel. Each curve shows the power
relative to the situation of uniformly-spaced robots aligned along the vessel
wall, i.e., the same normalization used in Fig. 26. The solid curves show the
average power for the 5 robots with average fluid speed $1\,\mbox{mm}/s$ and
$0.2\,\mbox{mm}/s$ for the upper and lower curves, respectively. The dashed
curves show the power available to the fifth robot for these fluid speeds.

Fig. 28 shows the average power for the robots and for the last robot as a
function of the offset angle $\theta$. The power is relative to the same
situation as used in Fig. 26. The nonuniform spacing has little effect on the
average power but aligned, nonuniformly spaced robots provide $20\%$ more
power to the last robot than uniformly spaced robots.

The choice of robot positions on the vessel wall alters the power available to
downstream robots in a manner similar to that of robots limiting their power
use, as discussed in Section 4.2, but without requiring the robots to
continually monitor and adjust their power consumption. Instead, robots
attaching to a vessel wall for extended operation could determine their
positions during their setup and then avoid devoting computation to
maintaining power limits while they perform their tasks while attached to the
vessel wall. On the other hand, actively adjusted power limits provide more
flexibility in distributing power to downstream robots. Robots could adopt a
hybrid approach of positioning themselves to best achieve their goals and
having upstream robots limit their power consumption when downstream robots
indicate they require more power.

As described in Section 3.3, robots moving passively with the blood and
attaching to capillary walls can lead to significant variations in the number
of robots in each capillary. Thus, in addition to adjusting their position in
individual capillaries, robots could alter how they divide among nearby
capillaries in a network of vessels. For example, robots could preferentially
position themselves in some of the capillaries in a network of vessels while
leaving others with few or no robots. In that case the increased vascular
resistance due to robots would tend to direct blood cells away from branches
of the network containing many robots, thereby contributing to the
heterogeneity of paths that cells take through a network of small vessels
[77]. Provided the open paths interleave closely with the vessels blocked by
robots, the flow through the open paths could be sufficient to support the
surrounding tissue with less disruption to the overall flow than if robots
were positioned in all the capillaries. The possibility of using just a
portion of capillaries to perfuse tissue arises because resting tissue can
contain more capillaries than required for adequate perfusion [25], though
with considerable variation among locations in the body [3].

###  4.4 Managing Heat Generation

Table 2 shows large numbers of robots add significant heat to the body. This
applies to any method robots use to generate power [29].

In terms of the chemical power discussed here, mitigating oxygen depletion by
transporting more oxygen (Section 4.1) does not address issues arising from
heating. Instead, providing more oxygen allows robots to increase their power
use, and hence produce more heat. Limiting power production can both reduce
oxygen depletion (Section 4.2) and reduce the heat generated by the robots.
Thus limiting power is the better approach if the large number of robots leads
to significant problems due to both oxygen depletion and heating.

Another method to mitigate heating is modifying the location-based power
limits discussed in Section 4.2 to change where the heating occurs. In
particular, robots could shift heat production to regions of the body that
readily dissipate the heat. Such regions might be detected by having a lower
temperature than the core of the body, such as near the skin. This approach to
heat mitigation is particularly suitable for missions where high power demands
occur near the skin, e.g., for robots communicating information to external
receivers when they are near the skin, such as described with the Markov
process example of Fig. 19. An alternative communication strategy of network
message passing through hubs located throughout the body would distribute
power use and heating throughout the body. With enough robots that heating
becomes an issue, heat dissipation could constrain the choice of communication
method, in addition to constraints on transmission rates, latency or
reliability.

##  5 Discussion

The circulation model developed here provides a quantitative assessment of
chemical power available to large numbers of microscopic robots in the
circulation. The simplifying assumptions used in this model suggest directions
for extending the model.

The model uses a fixed relationship between hemoglobin binding and oxygen
concentration in the plasma around the cell. However, hemoglobin binding
changes in response to other chemicals. In particular, elevated
$\mbox{CO}_{2}$ increases the release of oxygen by altering the cell
saturation equilibrium given in Eq. C.1. Carbon dioxide is a byproduct of full
oxidation of glucose. If fuel cells in the robots complete that oxidation, the
robots will increase $\mbox{CO}_{2}$. Accounting for this effect is
particularly relevant for a modest number of robots: few enough that cells
return to the lung with relatively high saturations but large enough to
noticeably decrease oxygen concentration. Since cells release more oxygen in
response to higher $\mbox{CO}_{2}$, they will provide those robots with
somewhat more power than estimated from the model used here. Alternatively,
robots might use a simpler fuel cell that only partially oxidizes glucose.
This can produce a similar amount of power as full oxidation because power is
limited by availability of oxygen rather than glucose [40], but would produce
partially oxidized glucose rather than $\mbox{CO}_{2}$ as a byproduct, and
thus would not have the same effect on hemoglobin binding.

The oxygen concentration used at the start of the circulation loop assumes
robots do not limit red blood cell oxygenation in lung capillaries. This is
reasonable for the number of robots considered here when they move passively
with the blood flow. However, if those robots remain in lung capillaries long
enough to fill oxygen tanks (see Section 4.1), the high concentration of
robots in lung capillaries could reduce oxygen that reaches red cells.
Adjusting for this effect requires evaluating the competition between cells
and nearby robots [40] and accounting for the kinetics of oxygen binding to
hemoglobin [16]. The result of this analysis would replace the initial plasma
oxygen concentration and cell saturation used by the model.

The model considered here follows a robot moving through an average
circulation loop. According to this model, a large number of robots consume
almost all the oxygen toward the end of the circuit, i.e., as the blood
returns to the heart in large veins. However, the actual circulation consists
of paths with a wide distribution of circulation times [60]. Blood from these
paths mixes in large veins before returning to the heart. Merging flows from
circulation paths of different lengths result in somewhat higher oxygen
concentration in large veins than suggested by the average-circuit model, and
the lowest concentrations will occur in long circulation loops just before
they merge with blood from shorter circuits. Extending the model to include
various circulation times is necessary to determine the minimal oxygen
concentration in the circulation due to robot consumption and where it occurs.

Another extension to this model is to consider robots operating outside blood
vessels. This requires extending robot oxygen consumption, discussed in
Appendix C.2, to also occur in the tissue around capillaries. Tissue has less
available oxygen than the blood [46, 63]. Moreover, oxygen in tissue, supplied
by diffusion from nearby vessels, takes longer to replenish from red blood
cells than when robots consume oxygen from plasma within vessels. Thus robots
operating in tissue have less available power and could lead to more
significant reduction in oxygen available to tissue than robots in blood
vessels. The extent of these effects depends on the number of robots that
leave vessels to operate in tissue and how long they remain. In addition,
variation in how rapidly oxygen diffuses from capillaries and in tissue power
demands in different parts of the body are not included in the average
circulation model considered here.

Robots using glucose fuel cells decrease oxygen in the body, add carbon
dioxide and produce heat. These changes could induce compensatory responses by
the body similar to those from higher tissue demand and heat production during
exercise, such as more rapid breathing and faster circulation speed.

How blood flow changes in response to robot oxygen use depends on how closely
robot consumption mimics the signals from increased tissue demand. For
instance, exercising increases metabolic waste products in active muscles
while robot consumption does not produce these waste products. One way to
estimate this response is to experimentally reduce oxygen directly rather than
as an indirect effect of increasing tissue demand [14]. The regulation of
blood flow through small vessels [64] could also be important in determining
this response. If such studies identify additional changes needed to induce
physiological responses to decreased oxygen, microscopic robot applications
could apply these signals, either directly by the robots or as an additional
intervention in conjunction with the robots.

The body may respond to robot consumption by increasing blood flow to tissues.
However, this would not bring additional oxygen to the same extent as the
normal response to higher tissue demand. This is because the additional blood
would itself contain oxygen-consuming robots. Thus, at the higher ranges of
robot numbers considered here and for long-term use, robots may need to
supplement the body’s response, for example by extracting additional oxygen
from the lungs into storage tanks. The body’s response and any supplementation
by robots could be included in the model through changes to the circulation
parameters, particularly the blood flow rate. More generally, the robots could
alter a variety of physiological properties. It will be important to include
these effects to create more accurate models [50] to guide the development and
use of these robots.

The fuel cells considered here use glucose as the fuel. This fuel is far more
abundant than oxygen [40], leading to the focus of this paper on how robots
affect the concentration of oxygen. However, full oxidation of glucose is a
complicated series of reactions [15]. Other fuels, such as hydrogen, have
simpler oxidation chemistry though are not normally available in the
circulation. If such fuels were added to the blood, e.g., via the lungs [17]
or injection into the blood, robots could use these fuels instead of glucose.
The distribution of these fuels in vessels could have significantly different
concentration profiles than that of glucose. In particular, fuel rather than
oxygen could be the limiting chemical is some parts of the body, requiring
extending the model considered here to evaluate the concentrations of both the
fuel and oxygen. In addition, a robot could contain fuel cells that react
different fuels, allowing it to select among fuels depending on their
availability. This capability leads to an additional mitigation strategy for a
swarm, whereby robots limit their power production from each fuel based on how
that affects available power to other robots in light of how those fuel
concentrations vary in the body.

As described in Section 2.2, the model discussed in this paper focuses on
steady-state oxygen concentration arising from robot consumption. In addition
to a steady-state reduction in oxygen throughout the blood, the timing of
robot activities may affect the body’s response. For instance, robots could
alternate between high-power active operation and low-power intervals where
they passively monitor for cellular changes or wait for instructions. If many
robots alternate at the same time, this could lead to large changes in oxygen
concentration. Due to delays between changes in oxygen consumption and the
body’s response, such variation can lead to instabilities in respiration over
periods of tens of seconds [27].

Instabilities arising from the timing of robot oxygen consumption contrast
with those that may arise even if robots use oxygen at a steady rate. For
example, large numbers of robots in the blood may alter vascular resistance,
which can lead to instability in blood pressure [44]. Another instability
could occur during extended missions, e.g., for health monitoring or treatment
evaluation over days or weeks. Since glucose and oxygen are continually
present in the body, their use for chemical power could support such missions.
In such cases, longer-term body responses could become important. One example
is if continual, repeated collisions between robots and blood cells
substantially reduce the lifetime of the cells, even though each individual
collision is unlikely to significantly damage a cell [29]. Due to delays in
production of new blood cells, such reduction in cell lifetime can lead to
instability in blood cell production [44]. The resulting oscillations in
hematocrit could affect oxygen available to tissue and robots.

Thus it will be important to evaluate temporal variation in physiology, at a
variety of time scales, from the use of large numbers of robots. Extending the
steady-state model discussed here to include variation in flow could help
estimate such effects.

##  6 Conclusion

This paper evaluates the effect of large numbers of microscopic robots using
chemical power as they move with the blood through a circulation loop with
average transit time. Normally, blood absorbs oxygen in the lungs and holds
that oxygen until reaching capillaries, where some of that oxygen diffuses
into surrounding tissue while the rest returns to the lungs. Robots, on the
other hand, could consume oxygen continually, not just during the short time
while the blood passes through capillaries. This leads to a continually
decreasing oxygen concentration throughout the circulation.

This analysis indicates that tens of billions of robots can each generate
hundreds of picowatts with relatively little effect on oxygen levels or
additional heating. As discussed in Section 3.2, this many robots are unlikely
to lead to detrimental consequences beyond local effects if and where robots
aggregate [40]. On the other hand, a trillion robots significantly reduce
oxygen, especially in the veins, and produce appreciable heating. If such a
large number of robots is necessary for an application, methods to reduce
these effects include providing additional oxygen in the blood or having
robots limit their power consumption. Section 4 applied the circulation model
developed in this paper to evaluate these mitigation strategies. As described
in Section 3.4, the model of robot performance and their systemic effects can
be adjusted to patient-specific parameters. More broadly, such models could
help develop individualized mission plans for microscopic robots.

Chemical power is one of several methods to power microscopic robots [29].
Acoustic power provided by transducers located on the skin is another option.
In that case, the main variation in available power is due to distance from
the skin and macroscopic acoustic properties, particularly attenuation, in the
tissue between the robot and the skin. Swarm behaviors can reduce the robots’
effect on attenuation [39]. For example, the high attenuation in lung
significantly reduces acoustic power available to robots in that organ. Thus
acoustic and chemical power are somewhat complementary: chemical power is
limited by oxygen concentration, which depends on the time since a robot last
passed through the lung, whereas acoustic is limited by attenuation and
distance from the skin. These power methods also differ in their
implementation: acoustic gives mechanical power whereas fuel cells give
electrical power, which needs to convert to mechanical to move robot
components, e.g., vibrating surfaces for communication or moving external
structures for locomotion. Thus selection among these or other power sources
will depend on the nature of the robots’ tasks, where in the body they require
the most power, and feasibility of design and fabrication of the power supply
within the robots. Large numbers of robots can significantly alter the
availability of alternate power sources in different ways. Thus accounting for
the effects of swarm behaviors is important in choosing how to power large
numbers of microscopic robots. The results of this paper indicate some of
these effects for chemical power.

## Acknowledgements

I thank Robert Freitas, David Lister, Ralph Merkle, James Ryley and Jeff
Semprebon for helpful comments.