© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

8.2.1.2 Arteriovenous Microcirculation

In most cases, the true capillaries do not directly join arterioles to venules. Rather, oxygenated blood passes out of the terminal branches of the arterioles (terminal arterioles are ~10-50 microns in diameter) into metarterioles (10-20 microns), "preference channels" whose walls contain smooth muscle fibers in declining numbers from proximal to distal, which eventually merge with the non-contractile postcapillary (8-30 micron) and collecting (10-50 micron) venules (Fig. 8.4). Capillaries branch off directly from the metarterioles through precapillary sphincters constructed of single muscle fibers; other than these, there are no contractile elements in the capillaries. The state of contraction of the sphincters controls the rate of fluid flow through the capillaries. In skeletal muscle, with its widely varying oxygen requirements, there are 8-10 capillaries per metarteriole; in the mesenteric circulation, a stable metabolism requires only 2-3 capillaries per metarteriole; in the nail bed, the ratio is 1:1.⁸³² The topology of the arterioles that supply the capillaries, and of the venules that drain them, is usually special for each tissue. Some beds are structured as trees, while others are organized into arcades, sinuses, or portal systems. Thus the vasculature of each organ is unique -- useful navigational information for medical nanorobots.

The wall of a capillary vessel is a tube consisting of a single layer of rolled-up endothelial cells. Endothelial cells may be less than 1 micron thick with a flat surface area of 300-1200 micron², so the entire ~313 meter² vascular surface may be comprised of up to 0.25-1 trillion endothelial cells. Endothelial cells are laterally apposed to each other with a 10-20 nm gap between neighboring cell membranes. These junctions are also somewhat organ-unique. There are three major types.

1. First, there is the continuous type with cells joined tightly together. Striated muscles may have flat thin cells, while in postcapillary venules the cells are cuboidal, forming a thick layer.

2. Second, there is the fenestrated type, with cells so thin that internal vesicles form small pores 25 nm thick and 100 nm in diameter (typically ~1000 pores/micron², compared to ~190 pores/micron² in arteriole endothelium and ~645 pores/micron² in postcapillary venule endothelium ).⁸³⁸ This type is found in the renal medulla, endocrine glands, and in structures engaged in the production or absorption of fluids such as the renal glomerulus, choroid plexus of the brain, and the intestinal villus.

3. Third, there is the discontinuous type with distinct intercellular gaps and a broken basement membrane, commonly found in sinusoid vessels and in organs such as the liver, spleen, and bone marrow whose functions include injection or extraction of whole cells, large molecules and extraneous particles from the blood.

Capillaries average 8 microns in diameter, but what is the minimum human capillary diameter? This question is important because it sets an upper limit on the size of bloodborne nanorobots. For example, in one experiment 97% of all 15-micron radiolabelled microspheres reaching the eye were trapped during the first pass.⁸⁴² Standard anatomy and physiology textbooks variously report that the minimum capillary lumen measures 5-7 microns,^834,839 4-9 microns,⁸³⁶ 4-8 microns,⁵¹⁷ or 4-6 microns^841,843 in diameter. There are also references to capillaries as narrow as 3 microns for laboratory rodents and other mammals.^834,840

Theoretical calculations^841,844 suggest that a cylindrical vessel must be at least 2.7 microns in diameter to allow maximally deforming human red cells normally averaging 7.82 microns in diameter ³⁶² to pass. This theoretical minimum assumes a mean cell (surface) area (MCA) of ~135 micron² and a mean cell volume (MCV) of ~94 micron³ for human erythrocytes, and allows for a maximum surface stretch of 10 micron² to a total of MCA ~ 145 micron².* From simple geometry for a red cell compressed into a hemisphere-capped cylinder during tube passage, the minimum tube diameter D_tube is given by

{Eqn. 8.1}

Experiments using polycarbonate sieves and micropipettes confirm that a tube diameter of at least 2.3 microns is necessary to avoid plugging.³⁷⁴

* Different species have red cells differing slightly in diameter, thickness, surface area and volume.⁸⁴⁰ For example, dogs have 7.2-micron red cells with MCA ~ 123 micron² and MCV ~ 69 micron³, giving (assuming surface elasticity characteristics similar to the human erythrocyte) a theoretical minimum passable tube diameter of 2.4 microns. Cats have 5.6-micron red cells with MCA ~ 83 micron² and MCV ~ 43 micron³, giving D_tube ~ 2.2 microns, a full 0.5 micron less than for human red cells.

However, the above theoretical computation assumes mean erythrocyte geometry and ignores the considerable dispersion in volume and surface area of individual physiological red cells around the mean value. Specifically, the largest 1% of human red cells have diameter d_RBC ~ 9.65 microns, MCA ~ 182 micron² and MCV ~ 132 micron³,³⁶² giving a theoretical minimum passable tube diameter D_tube ~ 3.1 microns for these cells. The largest cell in a population of 10⁸ cells (there are a total of ~300,000 of these largest cells in circulation at any given time, a not inconsiderable number) has d_RBC ~ 16 microns, MCA ~ 500 micron², MCV ~ 450 micron³,³⁶² giving D_tube ~ 3.7 microns. Thus if human capillaries were as small as 2.3-3.7 microns, a significant number of red cells would be unable to navigate passage and would quickly plug the tubes, resulting in angionecrosis. Experiments have also shown plugging by human leukocytes at tube apertures much below ~5 microns.^845,846 Thus the minimum viable human capillary diameter appears to be ~4 microns.

The average density of capillaries in human tissue is ~600/mm³,⁸³² which implies a mean separation of ~40 microns between adjacent capillaries averaging ~1 mm in length. Most living tissue cells lie within ~1-3 cell widths of a capillary. To reach a particular cell, bloodborne nanorobots may normally travel most of the distance through the vascular network, then exit the capillary and cross at most one or two cells to reach the desired target cell. Capillary density varies considerably in the vascular beds of different organs. For instance, microvascular density is 2500-3000/mm³ in the brain, kidneys, liver, and myocardium; 300-400/mm³ in phasic units of the skeletal musculature; and <100/mm³ in bone, fat, connective tissue, and in tonic units of the skeletal musculature.⁸³² Blood flow rate per gram of tissue (Table 8.4) is very roughly proportional to the relative microvascular density in each organ or tissue. More than 70% of the water of the blood is exchanged with extravascular water every minute -- the walls of smaller capillaries are veritable sieves with respect to water.²²²⁹

What is the size (in bits) of the minimum topological map needed to reliably navigate the entire ~19,000 kilometers of human blood vasculature? Map size is driven by the number of bifurcations (decision junctions; see Section 8.3.4) and capillaries. Taking the pulmonary arterial tree (Table 8.3) as representative, traveling from the heart to a final capillary requires passage through approximately 17 forks with an average of 3.2 branches per fork. In a simple bifurcation map with 17 forks and at most 4 branches per fork, each capillary can be assigned a unique topological address using 17 digits with each digit having log₂(4) = 2 bits, a 34-bit address vector. This specification is very compact because the minimum number of bits required to name each of 19 billion capillaries in the body is also log₂(19 billion vessels) ~ 34 bits. Using run-length coding, we can have 4¹⁷ addresses of length 34 bits, 4¹⁶ addresses of length 32 bits, and so forth, giving a whole-body minimum of: 34 bits (4¹⁷) + 32 bits (4¹⁶) + ... + 2 bits (4¹) = 763,549,741,512 bits for a single comprehensive bifurcation map.

Assuming some data redundancy (e.g., parity check bits) and an allowance for special cases (e.g., arterial anastomoses, shunts, or arteriovenous fistulas which are direct connections between a small artery and a small vein), a complete map of the human vasculature probably requires ~1 terabit to store the addresses of all 19 billion capillaries, with each address representing the unique navigational instructions needed to reach each vessel. This allows localization to within a ~50,000 micron³ volume or ~(40 micron),³ the approximate volume of each capillary. However, map stability at the lowest levels of the branching structure becomes a serious concern when the time interval between map compilation and map use exceeds 10⁵-10⁶ sec (1-12 days), due to angiogenesis and other continuous natural tissue remodeling activities, or in cases of tissue injury.

An arteriovenous vasculographic map might also provide an excellent framework for determining cytographic location in the body, since all tissue cells lie within a few diameters of a capillary and each capillary services only <~500 tissue cells. To make a complete cytographic map, the number of bits required to uniquely name each of the ~10¹³ tissue cells in the human body (Section 8.5.1) is log₂(10¹³ cells) ~ 43 bits; given similar allowances as before, <~1000 terabits are required to store the navigational address of every fixed cell in the body. Even using high-density hydrofluorocarbon memory tape it does not appear feasible for individual bloodborne nanorobots to carry a whole-body vasculographic or cytographic map onboard (the latter of which might require ~40,000 micron³ of data storage volume). However, a modest 0.1-micron³ 10⁹-bit map will allow a bloodborne nanorobot to retain the addresses (with complete navigational instructions) of up to 30 million individual capillaries (~50 cm³ of typical tissue) or ~20 million individual cells (~0.2 cm³ of cell-dense tissue) for the duration of its mission. Note that sequential passage through ~10 million different named capillaries of average length ~1 mm at a swimming speed of ~1 cm/sec (Section 9.4.2.6) requires ~12 days, a not unreasonable mission duration.

Last updated on 19 February 2003