A NASA manuscript dated 2006 (Sibonga and colleagues) compares dual-energy X-ray scans before and after flight for 45 crew members whose flights averaged 173 ± 24 days. CITED Bone mineral density at landing was below preflight by 6.8% at the femoral neck, 7.8% at the trochanter, 7.7% in the pelvis, 4.9% in the lumbar spine and 2.9% in the heel. The authors fitted an exponential recovery with a 50% recovery time for each site.
| Site | Loss at landing (%) | 50% recovery time (days) | 95% range (days) |
|---|---|---|---|
| Femoral neck | 6.8 | 211 | 129–346 |
| Trochanter | 7.8 | 255 | 173–377 |
| Pelvis | 7.7 | 97 | 56–168 |
| Lumbar spine | 4.9 | 151 | 72–315 |
| Calcaneus | 2.9 | 163 | 67–395 |
Their abstract says half the loss is recovered within nine months at every site, and that restoring fifteen sixteenths of it should take up to 36 months, with large differences between crew members. They add that restored density does not necessarily mean restored strength. The manuscript predates the Advanced Resistive Exercise Device, installed on the station in 2008, so these crews flew with earlier hardware; which device each of the 45 used was not checked.
A 2022 study in Scientific Reports (Gabel and colleagues) followed 17 astronauts with a three-dimensional scan of the shin bone and forearm before flight, at return, and 6 and 12 months later. CITED At 12 months, median shin-bone strength and several density and microstructure measures were still 0.9% to 2.1% below preflight; nine of the 17 had not fully recovered total density, and those nine had higher bone turnover markers. Missions longer than six months recovered worse: a strength deficit of 3.9% at 12 months, against 0.4% for shorter missions. The authors call this incomplete recovery, comparable to a decade or more of age-related loss on Earth. Only the abstract was read.
A 2012 NASA article on the Smith study reports that crews using the new device, with adequate energy and vitamin D, kept more bone density, and that crews before it lost 1 to 2% a month. CITED Its co-author Sibonga says it is still unclear whether the remodelled bone is as strong as before flight. The article, not the paper, was read.
The 2020 meta-analysis of bone loss in space travelers (Stavnichuk and colleagues) was found but its pages were blocked or empty, so no figure from it appears here. The 2012 paper itself and any numbers for crews on the newer device were also not read.
Frost’s mechanostat idea, as a reference page summarises it, is that bone adapts to the strain that muscle forces and gravity put on it. CITED Strain below about 800 microstrain is a disuse range in which bone mass and strength fall; between about 800 and 1500 the bone is adapted and holds steady; above about 1500 it adds bone. The page says these numbers are theoretical and differ by bone and person. Bone is then a feedback loop that corrects a deviation from a strain set point. The chain below is read from that loop, as a model; the spaceflight sources measure density and not strain.
Here h is the quantity the loop corrects (trabecular thickness or mass), L is the daily loading, ε0 the set-point strain, and r the strength with which a deviation is pulled back. MODEL
| Operator | Meaning in this room | Reading |
|---|---|---|
| C | reduction to one slow variable | The loop corrects one slow quantity, h, while strain on the bone is effectively instantaneous. The sources measure density at each site, which is one number per site. MODEL |
| K | curvature: how strongly a deviation is pulled back | K = V″ = r = ln 2 / (half-time). This one is sourced: 0.0033 per day at the femoral neck, 0.0027 at the trochanter, 0.0071 in the pelvis, 0.0046 in the lumbar spine, 0.0043 in the heel. The pelvis corrects deviations 2.2 times as fast as the femoral neck. CITED |
| F | the fold: K falls to zero at a critical value of the control | In this loop K = r at every loading, so the rate never vanishes as the load falls and there is no critical slowing. As modelled, F is absent. It would be present only if the half-time lengthened as the loading deficit deepened, toward a loading at which the correction stops. OPEN |
| U | relaxation to the new state, leaving a residue | After flight h relaxes back with rate K. The residue at 12 months is what one half-time predicts (section 3). A true residue needs something that cannot relax back, such as a lost strut that thickening cannot rebuild. That is a standard idea, not sourced here. OPEN |
The same thing in the project’s own words: K is the curvature, and in a fold the curvature at the critical point is zero, so the rate of correction goes to zero as the control approaches it. For bone the sourced K is a positive number at every site, and the question the chain asks is whether it shrinks with the depth of the loading deficit.
If K is constant, the half-time does not depend on how long the astronaut was away, and the deficit left at 12 months is proportional to the loss at landing. If K falls with the depth of the deficit, the residue grows faster than the loss. The 2022 abstract reports a 12-month strength deficit of 0.4% for missions under six months and 3.9% for longer ones, a factor of about 10. Whether that is proportional depends on the mean durations of the two groups and on whether loss accrues evenly, and neither number is given. With per-crew landing losses and half-times against mission length, the test is one regression: a half-time that lengthens with mission length is the signature of K falling, and a flat one says F is absent.
There is also a driver threshold in the loop, separate from K: loading below Lc = ε0 hc would thin the struts past a critical thickness hc if flight lasted longer than Tc = (1/r) ln[(hE − h*)/(hc − h*)], a duration that grows without bound as loading rises to Lc. That is the part of the chain where a mission-length threshold such as the six-month split could come from. hc and the perforation law are assumptions with no data here, so no value is claimed. OPEN
The dm3 table gives each room a rate μ and a frequency ω, so that near the threshold the room looks like μ ± iω. A fitted exponential recovery has no frequency: it is one real eigenvalue, μ = −ln 2 / (half-time). That gives, per day, −0.0033 at the femoral neck, −0.0027 at the trochanter, −0.0071 in the pelvis, −0.0046 in the lumbar spine and −0.0043 in the heel. Written per month, the femoral neck is −0.10. The confidence ranges are wide: the femoral-neck half-time runs from 129 to 346 days, a factor of 2.7. These are the nearest thing to a sourced μ that any room in the Chapter 20 table has; no value in that table has a source in the pages searched.
With ω = 0 the bone row is not a spiral sink, which needs μ < 0 and ω > 0. The immune chapter has the same mismatch from the other side: a fold has a real zero eigenvalue, not a complex pair. So the table’s shared claim, that every row is a spiral sink, does not extend to bone as fitted.
The residue is also not a sign of a fold. A single half-time of 211 days leaves 0.5365/211 = 30% of the landing loss after a year. Applied to the 2006 landing losses, that predicts a 12-month deficit of 2.05% at the femoral neck, 2.89% at the trochanter, 0.92% in the lumbar spine, 0.61% in the heel and 0.57% in the pelvis. The 2022 shin-bone deficits of 0.9% to 2.1% overlap two of those, but they are a different bone, measured a different way, in different crews, so this is a consistency check and not a test. It does show that “not fully recovered at one year” is what ordinary slow recovery predicts.
6.8% over 173 days is 5.68 months, which is 1.2% per month if loss accrues evenly (1.14% using the 6.5% printed in one figure caption). Both fall inside the “1 to 2 percent per month” of the 2012 article. The verify script reproduces this and every other number in this section.
Chapter 4 of this book says plainly that its κ* values are stated per domain with no computation from the domain’s geometry, and are to be treated as fitted proxies awaiting calibration; Chapter 7 says they are derived from geometry. No page gives a definition of the curvature κ for bone. So there is no κ* for bone to test, and this chapter proposes none. Inventing one now would be the move the V5 cut reversed.
Pre-stated, so the answer can go against the framework. If bone belonged in the table with a threshold, the data would show a driver (mission length, or loading per day) with a critical value where the recovery rate vanishes like the square root of the distance and a different path back follows, and some quantity a reviewer could name as κ would cross a value fixed beforehand. The sources show instead a single exponential with half-times from 97 to 255 days and no sweep. The six-month contrast in the 2022 abstract (0.4% against 3.9%) is two groups; it cannot tell a step from a slope, and the mean durations of the groups are not given.
Two tests the data could support, if the raw scans or per-crew values were held: whether each crew member’s recovery is a single exponential, and whether the 12-month deficit grows with mission length in proportion or faster than proportion. Neither is run here.
If a bone row is wanted in the table, the entry today is: C a modelled single slow variable, K = r sourced per site, F absent as modelled and testable by half-time against mission length, U a residue consistent with one relaxation, normal form a real-eigenvalue relaxation (not a spiral sink), κ* not testable. To go further, the data to find are crew-level recovery curves on the newer device, the 2020 meta-analysis, and any experiment that varies loading dose and records recovery rate near a threshold. A toy remodelling model with a threshold could be hand-derived and run through the fold test as was done for allostatic load, but it would be a model of the framework, not a statement about bone.
Whether elastic bands with paced loading could replace a compact resistive device is not settled by anything here: none of the sources tests bands.