© 2003 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume IIA: Biocompatibility, Landes Bioscience, Georgetown, TX, 2003

15.3.8 Nanorobotic Thermocompatibility

The issue of nanorobotic thermocompatibility arises in at least two contexts: First, the active production of waste heat (or localized cooling) by individual nanorobots, nanoorgans, or other nanorobotic systems implanted within the human body; and, second, the physiological effects of nanomedical implants that may result from the passive thermophysical characteristics of those implants, or from the materials with which they are constructed.

Previous discussions of thermally active systems include nanorobot waste heat conduction (Section 4.6.6), the local and global in vivo thermogenic limits of nanorobotic systems (Section 6.5.2), thermographic navigation (Section 8.4.1), and the thermal safety of in vivo electrical (Section 6.4.3.1) and mechanical (Section 6.4.3.4) systems. For instance, excessive nanorobotic waste heat generation (e.g., creating localized temperatures >42 ^oC) can stimulate thermosensitive channels in keratinocytes and in a specialized group of heat-sensing sensory neurons terminating in the skin [5669].

Previous discussions of passive conductivity include the thermophysical characteristics (Table 8.12 and Appendix A) and the thermal conductivity (Section 10.5.4) of biological and other materials. Individual nanorobots also can alter the thermophysical properties of biological tissues or fluids, although typical therapeutic terabot doses [2762, 3573] should not produce clinically significant effects. Maxwell’s theoretical model [5629] predicts that the effective thermal conductivity K_eff of liquids containing suspended micron-size (and larger) spherical particles increases with the volume fraction f_nano of the suspended particles as: K_eff/K_liq = 1 + {3 f_nano / ([(K_nano + 2K_liq)/( K_nano - K_liq)] – f_nano)}, where K_liq is the thermal conductivity of the pure liquid, K_nano is the thermal conductivity of pure particles (e.g., nanorobots), assuming the particles do not interact thermally (e.g., f_nano<< 1) [5630]. The interaction between spheres even at large volume fraction, as calculated by Rayleigh, produces only a small correction, which is why Maxwell’s simpler derivation is usually employed [5630]. For “perfectly conducting” (i.e., infinite K_nano) spherical particles, Maxwell’s model further simplifies to K_eff/K_liq = 1 + {3 f_nano / (1 – f_nano)}; Lu and Kim [5631] found that perfectly conducting prolate spheroids with a/b = 10 (length/width) give theoretical and experimental values at least 10% lower than the K_eff/K_liq for spheres.

Thus for example, in the case of a ~1 cm³ dose of spherical diamond nanorobots (~10¹² devices at ~1 micron³ each) infused into the 5400 cm³ adult human male blood volume (f_nano = 1/5400), taking K_nano = 2000 W/m-K for diamond and K_liq = 0.549 W/m-K (whole blood; Table 8.12), then K_eff/K_liq = 1.00056, which is clinically insignificant. Blood heat capacity similarly is virtually unchanged. The conclusions are much the same for soft tissue emplacements of similar nanorobot populations. In augmentation scenarios [3573, 4609] (Chapter 30) involving heavy loadings of the human body with diamondoid particles, the results are not much different. At 37 ^oC and a maximum 10% nanocrit (Section 15.6.2), blood heat capacity falls only 5%, from 3.82 MJ/m³-K to 3.62 MJ/m³-K for pure diamond particles, while blood thermal conductivity rises only 33%, from 0.549 W/m-K to 0.732 W/m-K for pure diamond particles – somewhat less than the conductivity of live brain tissue (Table 8.12), hence is probably not clinically significant.* Similarly, thermal equilibration time is only modestly increased even in the case of ~1000-terabot augmentation loads of free-floating in sanguo nanorobots.

* Most studies of the thermal conductivity of suspensions have been confined to those containing millimeter- or micron-sized (i.e., nanorobot-sized) particles, but nanometer-sized particles have a still larger surface area-to-volume ratio and thus might be expected to exhibit higher thermal conductivity because heat transfer takes place at the surface of the particle [6003]. Accordingly, experiments on “nanofluids” by Choi and Eastman [6000-6004] revealed that even a small volume fraction f_nano = 0.003 (0.3%) of 10-nm metallic copper particles suspended in ethylene glycol produced a 40% increase in thermal conductivity of the composite fluid (i.e., K_eff/K_liq = 1.4) – roughly an order of magnitude larger increase in conductivity than the classical Maxwell theory predicts. Alumina-particle nanofluids have also been investigated experimentally [6005].

Nevertheless, changes in whole-body thermal conductivity due to massive implantation of continuous diamondoid materials can impact natural thermoregulatory mechanisms. Aside from blackbody radiation, sweating, capillary sphincter control, and behavioral thermoregulation (including respiratory cooling), the body regulates its temperature and offloads excess heat principally via two mechanisms, as follows.

First, there is passive conduction. Heat travels by pure conduction through fat and muscle from the body core out to the periphery. The thermal conductivity of human tissue is K_t ~ 0.5 W/m-K, so for a typical L = 10 cm path length (~half-torso thickness), heat flow H_f ~ K_t / L = 5 W/m²-K, or ~10 W/K for a 2 m² human body. In a cold room, the mean temperature differential between core and periphery DeltaT ~ 11 K (Section 8.4.1.1), so H_f ~ 100 watts, which is approximately the basal metabolic rate. Experiments confirm that 5-9 W/m²-K is the minimum heat flow in very cold conditions (the actual value depending largely upon the thickness of subcutaneous fat layers) [2093]. In this case, the peripheral capillary blood flow has slowed to a trickle, producing the minimum thermal conductivity of the human body in cold conditions. On the other hand, in a warm room or during heavy exercise, DeltaT ~ 1 K, so H_f ~ 10 watts. Thus, paradoxically, at warmer temperatures when the human body is generating considerable surplus heat, the body’s passive heat flow is actually very low because of the smaller temperature differential between core and periphery.

Second, heat is transported via the active blood flow. In warm rooms, not only are the peripheral capillary sphincters fully dilated, allowing more blood to flow through the peripheral capillaries relative to the core capillaries, but also the total volume of blood flow may increase. (During heavy exercise, total blood flow volume may rise by a factor of 4 or 5.) Diathermy experiments suggest that the active blood flow mechanism alone may carry off 100-200 watts of heat before core temperature starts to rise (Section 6.5.2). In cold rooms and in the absence of heavy exercise, peripheral capillary sphincters are maximally contracted, thus minimizing blood flow (and hence heat transport) to the periphery.

To summarize: The passive conduction mechanism can throw off ~100 watts of waste heat when the human body is in a cold room but only ~10 watts when the body is in a warm room, while the active conduction mechanism can throw off negligible heat in a cold room but up to 100-200 watts in a warm room. Thus as the external environment warms up, the human body shifts from passive conduction to active conduction via increased blood flow and capillary sphincter widening.

The presence of even a maximum 10% Nct of diamondoid nanorobots in the circulation will not significantly alter the heat capacity of the blood, hence the active conduction mechanism in human thermoregulation should be largely unaffected. However, in augmentation situations where vascular fluid flow is completely replaced by nonfluidic transport systems (and including capillary sphincter inhibition) as in the vasculoid [4609] (a whole-body diamond-plated artificial vasculature; Chapter 30), the active conduction mechanism in thermoregulation is essentially disabled.

This leaves the passive conduction system. Heat flow in a natural biological-tissue body is H_f,biol = K_t / L = 5 watts/m²-K. For a human body shape composed entirely of pure diamond (K_t ~ 2000 watts/m-K at 310 K [5632]) and again taking L = 10 cm, then H_f,diam = 20,000 watts/m²-K. For a diamond-envasculoided human body, taking a mass of ~1.7 kg of diamond thoroughly interwoven with 68.3 kg of mostly aqueous biological tissue mass (for a standard 70 kg male body), as a worst-case estimate* the effective heat flow becomes H_f,vasc ~ (1.7 / 70) H_f,diam ~ 500 watts/m²-K, or (H_f,vasc / H_f,biol) ~ 100 times more thermally conductive than before. For comparison, a pure metal human form would have H_f,metal ~ 170 watts/m²-K for stainless steel, ~350 watts/m²-K for lead, ~780 watts/m²-K for iron, or ~3800 watts/m²-K for copper. Hence a diamond-envasculoided or augmentation-loaded human body, in the worst case,* could have passive conduction properties similar to those of solid metal.

* This calculation pessimistically assumes that the continuous diamond implant is proportionally represented in conductive channels oriented normal to the body surface. B. Wowk notes that one can also consider the opposite extreme in which the diamond implant is wholly oriented as parallel rods running through tissue parallel to the body surface. The thermal conductivity enhancement between body and external environment would then be much smaller, for the same reason that highly conductive particles that occupy a small volume fraction in solids or liquids don’t significantly enhance conductivity (see above). The reality of an artificial diamondoid vasculature would lie somewhere between these two extremes. A proper quantitative thermophysical assessment of the vasculoid [4609] would require an analysis of the effect of fractal tortuosity on thermal conductivity calculations. Wowk notes that evacuated aerogels demonstrate phenomenally low conductivities (0.005-0.01 W/m-K) [6006] for their glass content and their study might provide some rough quantitative guidance on the effect of fractal tortuosity on thermal conductivity, but a more accurate result will require thermal modeling on numerically-generated random vascular trees. Mathematical models of thermal tortuosity have been explored in other contexts [6008, 6009], and fractal tortuosity has found its way into the reaction chemistry [6007], fluidized-bed engineering [6008-6013] and hydrogeology [6014] literatures, but fractal thermal tortuosity apparently has not yet been extensively studied.

This has implications for the maximum DeltaT that can be maintained between core and periphery. Consider a human-shaped tissue-mass with half-thickness L ~ 10 cm and surface area A ~ 2 m², sufficiently heated from the inside to cause P ~ 100 watts (human basal rate) to flow via passive conduction from core to periphery, establishing a temperature differential DeltaT ~ P L / A K_t ~ 10 K for natural human tissue with K_t = 0.5 watts/m-K. Upon switching to diamondoid-envasculoided tissue, mean tissue thermal conductivity would rise to K_t ~ 50 watts/m-K and so DeltaT would fall to ~0.1 K. In effect, the entire human body would become isothermal to within 100 millikelvins; even at the peak power output of 1600 watts for the human body (Table 6.8), DeltaT rises to just ~1.6 K. Thus a diamond-envasculoided human body would tend to become isothermal with its surroundings very quickly (although partially resisted by intervening subcutaneous fat), a possible hazard to normal human health especially in very hot or very cold environments. The thermal equilibration time is approximately t_EQ ~ L / v_thermal ~ 0.1 millisec, where v_thermal ~ K_t / h_plate C_V = 1000 m/sec for neighboring vasculoid plates in good thermal contact with each other and having thickness h_plate ~ 1 micron, with K_t = 2000 watts/m-K and C_V = 1.8 x 10⁶ joules/m³-K for diamond at 310 K, and taking L = 10 cm as before. This is far shorter than the typical 1-10 sec thermal response time of the purely-biological human vasculature.

The substitution of sapphire for diamond in these augmentation applications should significantly improve thermal performance. The thermal conductivity of synthetic sapphire may be as low as K_t ~ 2.3 watts/m-K for sapphire at 310 K [5633], roughly a thousandfold lower than for diamond, when measured in the direction normal to the symmetry or optic axis (c-axis); heat capacity (C_V = 2.9 x 10⁶ J/m³-K) and density (3970 kg/m³) of sapphire are slightly higher than for diamond. Thus for a sapphire-envasculoided human body, taking a mass of ~1.9 kg of sapphire (at 25 watts/m²-K for L = 10 cm) thoroughly interwoven with 68.1 kg of mostly aqueous biological tissue mass (at 5 watts/m²-K), the total heat flow is just H_f,sapph ~ 5.5 watts/m²-K, which differs insignificantly from natural biological tissue [5635]. At P = 100 watts, DeltaT falls to 2 K compared to 10 K for natural tissue and 0.1 K for diamond-envasculoided tissue; t_EQ ~ 1 sec for sapphire vs. 10^-4 sec for diamond.

Two complications regarding sapphire require additional research. First, the thermal conductivity of sapphire can vary significantly with both composition and crystallographic orientation, a fact which may impose additional and unknown constraints on the various designs. For instance, one source reports heat flow values interpolated to 310 K of 21 watts/m-K normal to the c-axis and 23 watts/m-K parallel to the c-axis [5632]; minor extrapolations of other sources to 310 K (i.e. slightly outside the exact temperature ranges measured experimentally) imply values of 2.0 [5634] and 2.3 [5633] watts/m-K for heat flows normal to the c-axis. However, all reported values for sapphire are at least two orders of magnitude more insulating than diamond.

Second, much like diamond, the thermal conductivity of sapphire varies with temperature. For example, at ~200 K (near dry ice temperature) sapphire’s thermal conductivity rises to 5 watts/m-K. At liquid nitrogen temperature (77 K), K_t soars to ~1000 watts/m-K; the peak is ~6000 watts/m-K at 35 K [5632-5634]. (Diamond’s conductivity also rises as it cools [5632-5634].) At the other temperature extreme, sapphire’s thermal conductivity rises to 3.9 watts/m-K by 523 K. Diamond thermal conductivity also varies significantly with isotopic composition (e.g., ¹²C vs. ¹³C [5636, 5637], Table 4.1); in 2002, it was unknown whether similar opportunities might exist for the engineering of desired levels or patterns of thermal conductivity in isotopically-controlled sapphire-based nanorobotic devices.

Last updated on 30 April 2004