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## Proper Vitamin Dosage

Let’s get the math right first

#### The Big Idea

It would be too much to ask of you, my gentle browser, to task your patience by expecting you to read this entire page without a clear idea of what you are getting yourself in for.

I picture myself being limited to convey the Big Idea Here in a text message or two. This is my humble attempt to convince you to read on…

Imagine that you are an alien scientist, sent to planet Earth to discover clues to the mystery of Vitamin C: why most mammals can make it in great volumes from blood sugar; why primates, some bats and guinea pigs lost that ability but consume large volumes in diet each day; and why the primates called homo sapiens have come to consume so little.

You realize that the true mission of Vitamin C is to protect and defend, and to repair and replace, but this is part of a broader cycle of metabolic processes. The cyclical pattern of Vitamin C’s density in the blood is exactly consistent with this metabolic pattern.

This daily cycle is the catabolic (break down) phase during exertion and the anabolic (build up) phase during rest. Vitamin C usage would also fit into an annual cycle of summer and winter behavior patterns.

# Executive Summary

To say that this page is “technically heavy” is a bit of an understatement. Before diving in, it’s worth offering a very very big picture of what lies ahead.

I want to contrast two competing contexts in terms of which to consider the question of micronutrient dosage in general, and of Vitamin C in particular (as a proxy of a broad range of micronutrients). The first context is the science of inanimate objects; the second context is the science of living things.

The first context might be simplified as the image of a bucket, full of water draining out of an open spigot or hole. In the case of the bucket, the water continues to drain out until a stable condition is reached: when the level of the water is below the spigot or hole. There is a mathematics for how this inanimate system gets to a stable condition; it’s called a relaxation process. Most medical science considers this model to be an accurate representation of how soluble vitamin levels works in humans.

The second context might be simplified as the image of a hummingbird floating in front of a flower. The hummingbird floats there in a stable position because it intends to. This stability is dynamical and kinetic and intentional. It continues only because the hummingbird works at it.

Using the first context (the science of inanimate objects), medical science suggest that it is pointless to take more than a certain level of Vitamin C since it will just leak away, literally, like water from a bucket, charge from a battery, 0r heat from a cup of tea. This is like saying one should not eat carbohydrates because the heat they produce within your body will just leak away.

However, micronutrients have a particular mission in the human body: to protect and defend against accumulated damage. The degree of that damage can be estimated as a function of the age of a person with a well-known curve (Gompertz-Makeham). It shows that accumulated damage increases exponentially after the age of 20 years. It is appropriate to estimate the need for micronutrients in proportion to the expected accumulated damage that they would be employed to repair.

If you had the job of repairing homes in a neighborhood after a hail storm, you would try to get an estimate of how much of what materials were necessary to bring to the job site. How many roofing tiles of what types? How much window glass of what sizes? How much paint of what colors? More damage would mean more repair materials. So it is with something like Vitamin C. In the absence of other information (like a diagnosis of a specific illness) the best information about accumulated damage comes from a person’s age and the Gompertz-Makeham curve.

Using the concept of dynamic kinetic stability in living things and estimating the need for micronutrients in terms of the Gompertz-Makeham curve, a progressive dosage of Vitamin C can be suggested that elevates the daily levels of the micronutrient in proportion to the expected accumulated damage within the individual.

Do not get bogged down with the technology in what follows. Skip past it if it’s too heavy to digest in a single sitting. The core concept is simple enough. We are living creatures that need micronutrients to protect and defend against specific kinds of tissue damage. The greater the degenerative damage that we’ve accumulated, the greater our need for repair and regeneration.

# Proper Vitamin Dosage

## Imagine

Imagine that you are a space alien scientist traveling the universe, and you come upon planet Earth 20,000 years ago. As you begin your studies of life on the planet, you discover the mammalian life and become interested. As you study the metabolisms of these creatures, you learn that there is a molecule ($C_ 6 H_ 8 O_ 6$) that almost all of them can synthesize from a sugar in their blood ($C_ 6 H_ {12} O_ 6$). This molecule is essential to many of the metabolic functions of these animals, from the creation of their tissues (collagen) to regulation of fat storage to production of neurotransmitters and more. In a typical mammal with a mass of 70 kg, 4 grams of this chemical is made each day; and under conditions of stress, 13 grams may be made.

Yet, there is an odd group of mammals that have lost the ability to make this chemical. There a little guinea pigs. There are several kinds of bats. And there are many primates, especially the great apes. They all have fruit in their diets. For some of these animals, fruit may make up 80% or more of their diets. Fruit is a fine source of this critical chemical. A gorilla with a mass of 70 kg typically gets as much as 4 grams of this chemical each day from the fruit in its diet. In short, the creatures that have lost the ability to synthesize this critical chemical get plenty of it from their diets anyway.

Except for one, the homo sapiens creatures, the ones that have developed language and tools and are beginning to spread out of Africa and to other parts of the globe. Your explorations have shown you that these homo sapiens used to have a diet very high in fruit as well. Now, as they have moved away from the rich jungles of Africa, their diets have changed. They cannot get as much fruit as they once had easy access to. On occasion, they will even develop a deficiency of this vital chemical. Their teeth will fall out, their gums will bleed, they become irritable, eventually they will even die. They can be otherwise well fed, and yet, they’ll die without enough of this molecule. You do some measurements and find out that they still have around 300 mg of $C_ 6 H_ 8 O_ 6$ when they sicken. Depending on their size, and some can be quite large, they might have anywhere from 1.5 gm to 4 gm or more in total.

You wonder what advantage their history could have bestowed on these homo sapiens creatures in becoming unable to make this vital chemical from that sugar in their blood? But your time is up, and so you head back home across the vast expanses of interstellar space.

You make your report on your home planet and that’s that. But your peoples’ memories are long and the mystery of the primates, bats and guinea pigs of planet Earth comes up again thousands of years later. A computer intelligence spits out the question to the scientific community, and folks begin to wonder what has become of those homo sapiens who were spreading out of Africa 20,000 years ago. So a new space ship is sent out with the mission of finding out where things stand. The life-spans on your home world are so long, and time being odd for trips at near light speed, that you are on this expedition to planet Earth once more.

When you arrive, just yesterday, you find that those homo sapiens creature have more than succeeded in leaving Africa; they now have over-run the planet. They have their own science, and they too have wondered about this critical chemical, which they call “Vitamin C”. They have found what you found some 20,000 years earlier; namely, that they will sicken and eventually die if they do not get enough of this in their diets. They call that sickness “scurvy” and an individual who is getting scurvy is called “scorbutic”.  They also call Vitamin C “ascorbic acid” meaning that it is “anti-scurvy”. After finding that diets that do not include enough ascorbic acid cause scurvy several hundred years earlier (a tick of time on your world), they have begun to take tablets that contain Vitamin C. Very few of these homo sapiens fellows eat anything like the diets that their ancestors consumed. In the more well-off parts of the world, even though a variety of food-stuff is available, most of them shun fruits and vegetables in consume large volumes of highly processed and micro-nutrient poor food. The agricultural industry is growing and delivering fruits and vegetables that are much poorer in vitamins and minerals than the same stuff that grows in the wild. In poorer parts of the world, diet is more limited and the focus tends to be on some staple cereal crop that can sustain life, but that does not optimize it. Micro-nutrient deficiency diseases are more common than a scientist would care to see, overall.

They too have discovered that most other mammals can make Vitamin C from that blood sugar, which they call glucose. They call the enzyme that the other mammals have and that they do not have, L-gulonolactone oxidase.

But there is a controversy among the homo sapiens creatures: how much of this Vitamin C stuff is the right amount? Some say that they should be taking as much as the other mammals that can make it themselves produce, or that their other large ape cousins obtain in the wild: several grams a day. Others scoff and say that a few 10s of milligrams is more than enough. They add that if more than 200 mg/day is consumed, the excess will simply be swept out of the blood stream by a normal function of the kidneys anyway. You decide to check this for yourself: you transport up several homo sapiens from around the planet and run some experiments on them. You give them large doses of ascorbic acid straight into their blood streams and you find that indeed, almost all of it comes out in their urine fairly quickly, within a few hours anyway. You begin to wonder if their doctors who think that high doses of the vitamin are pointless aren’t correct. And yet, you wonder, what makes these homo sapiens creatures so unique on the planet that their Vitamin C requirements have become so much less than any other mammal their size?

You begin to attack the mystery of Vitamin C. You beam up some computers and tap into the Internet and start scanning with the “Google” and the “Wikipedia”.

## Why is it so difficult to work out a daily dose?

It seems that everywhere you look, you see conflicting information about how much of any vitamin is the “right daily dose”. You find that the RDA among the homo sapiens used to be 60 mg and now it’s 90 mg. You find that some NIH researchers are suggesting that should go up to 200mg. One notable homo sapiens, Linus Pauling, was suggesting 1,000 mg (1 gm) and some folks recommend even higher daily doses. Oddly enough, you find this web site and learn that I myself am taking nearly 1,500 mg on a daily basis, so you can tell where I come down on the issue.

That original RDA of 60 mg was established to deliver enough Vitamin C to prevent scurvy. Scurvy is what happens when you no longer have enough Vitamin C in your system to maintain collagen, the binding goop that keeps your tissues together. When that happens, a primate’s teeth fall out, they start to bleed; and they might as well have that Ebola virus or some other hemorrhagic fever. More recent RDA values have been based on other factors concerning the body’s use of Vitamin C. A new proposal for an RDA of 200 mg is based on research in healthy individuals (all young male homo sapiens) that involved monitoring the uptake of Vitamin C in certain cells in the immune system, as well as plasma levels. It becomes apparent to you that homo sapiens science is recognizing that Vitamin C plays a role in multiple internal systems. That 200 mg target is not simply based upon an avoidance of the scurvy deficiency condition, but also considers optimizing the performance of an essential body function; viz., immunity. It seems that the notable Linus Pauling is having some impact on modern medical thought among the homo sapiens scientists.

Yet even here there is a strange discrepancy. Some of these homo sapiens doctors care for other animals, including other large primates, in places they call zoos. These animal doctors have studied the biology of the other primates in some detail, and they have published books and more books and papers on how much Vitamin C should be fed to gorillas, orangutans, chimpanzees, and so on. Sure as anything, the animals doctors recommend feeding more Vitamin C to other large apes in captivity and suggest the same for homo sapiens diets.

It seems that only the human doctors specializing in humans are the ones saying that smaller daily doses are the right way to go.

# Homo sapiens are buckets

You find that the human human doctors have a model for how Vitamin C works in the blood stream. The basic idea is that a human is like a bucket with a spigot part way up. You make a quick drawing of a human as a bucket:

The Human Bucket

If you fill a bucket with water up to the top and flip open the spigot mounted part way up the side, then the water starts to run out. The water rushes out at the beginning and trickles out as the level falls to where the spigot is. This is such a broadly known phenomenon that it has a special name in science; namely, relaxation. Relaxation is what happens to some system that has been perturbed (disturbed or modified) as it gradually comes back to normal. A bucket with a spigot open is in a steady state only when the water level is at the spigot level. If you start it full, then water is going to keep running out until it gets to where the spigot is. Then it will stay at that level forever unless you try to put some more water into it.

Other things do that relaxation trick. Charge up a battery and connect it to some electrical circuit and the battery will discharge until it’s empty, just like the bucket emptied itself of water. And so on and so on.

The homo sapiens doctors’ model of how their bodies lose a micro-nutrient like Vitamin C is based on a relaxation model like this. It assumes that if a human body started with some big level of Vitamin C in the blood stream, then the way in which that Vitamin C would leave the body would be well-approximated by the same model of how water drains out of a bucket or of how electrical charge drains out of a battery.

The rate of drop in water level is proportional to the water level

## Half-Life

There’s a way to think about how long it takes the water to run out of the bucket. You think about how long it takes to drop by one-half. This half-life idea shows up in many situations in science. It’s how Carbon-14 is used to date the age of organic material, like old bones. Here we have the idea of the half-life of the water in the bucket. It turns out that $\tau _{\frac{1}{2}}=\tau \ln (2)$. In that little equation $\tau _{\frac{1}{2}}$ means the half-life. Also, $\ln(2)$ is the natural logarithm of 2, and that’s around 0.693….

Suppose we start with the bucket level at 1 meter and the spigot right at the bottom. At the half-life time, the water level is down to ½ meter. Wait another half-life time; and it’s at ¼ meter. Wait another half-life, the level is ⅛ meter, and so on. This is useful knowledge. Suppose the half-life was 10 seconds. Then you know that at 40 seconds, the level will be 1/16 of where it began, or very little.

## Back to Vitamin C

The human human doctors figure that this is how Vitamin C leaves the human body; that it has a given half-life. They figure that it is the kidneys that are sweeping Vitamin C out of the blood stream. If you put a large amount into the blood, then the kidneys will just filter it out very quickly until only a relatively small amount remains. They call this amount “renal threshold”. It is like having the spigot of the bucket part way up.

In other words, putting a lot of Vitamin C into a human’s blood stream is like filling up a bucket with water when the bucket has an open spigot on the side wall somewhere. The water all flows out until the level is down to where the spigot is. Same thing for Vitamin C: take a big dose of the stuff and the kidneys drain it all away until the level gets down to “renal threshold”.

### Some pictures of all this

You make yourself a quick picture of the human human doctors’ idea [Click on the thumbnail for a larger version. Click outside to come back]:

Exponential relaxation

You can see that we start the clock at 0 and water gets to draining. The units of time are $\tau$, whatever that is. By t=0.69 or so, the bucket is now only half full; this is our first half-life gone. At t=1, the level is down to 36.7%. When the time is up to just less than 1.4, the bucket is now ¼ full. That’s two half-lives. And when the time is not quite 2.1, or three half-lives, we’ve only got 1/8th of a bucket left. And by the time we get out to t=5, there’s less than 1% left.

The human human doctors say that this is the way that Vitamin C drains out of people. Instead of going right down to zero, it goes to “renal threshold”, but still. If anyone asks them about taking large doses of Vitamin C, they don’t try to explain all of this machinery. Instead they just say, usually in a frustrated manner, they’re wasting their hard earned cash on making expensive urine.

If some human were taking a good sized dose of Vitamin C each morning and also that the human human doctors were right and the half-life of Vitamin C is only 30 minutes, then you sketch out this picture of what their Vitamin C level looks like over several days:

The doctors’ view of vitamins

Your quick sketch doesn’t have the exact units here properly. A human human doctor would likely want to see your graph in terms of some unit of the mass of Vitamin C relative to blood plasma volume; but this is about right. The units of time are days, and it shows what might happen with large doses of Vitamin C on a once-daily basis. You average this out, and you find that the average is only 1% or so of the maximum. Maybe the human human doctors are right: humans that take high doses of Vitamin C are simply leaking this stuff out so fast that it’s not worth bothering with.

But you sit back even more mystified. You have this one creature on this planet that cannot make its own Vitamin C. In evolutionary terms, its ancestors seem to have lost the ability to make it because they just didn’t need to any more: they were getting more than enough from their diets. But now this homo sapiens creature is not getting anything like the original amount from diet, they don’t store all that much; and in fact, it seems if they take higher doses, instead of storing more, they just leak it all away as fast as they possibly can.

The whole story just does not hang together in any logical way. You begin to think Vitamin C is a total mystery.

You know that there is a lot wrong with your simple curve. You figure that human tissues are more complex than a bucket with a spigot. You’re aware of different relaxation models than this easy one. Lots of systems exist with whole broad ranges of internal components that relax with whole broad ranges of time constants. You bet that humans are more like that than this simple bucket model.

## What the doctors know is wrong already

The human human doctors are already aware that the bucket model has some issues. First of all, you find that they realize that a 30 minute half-life applies only to very high doses of Vitamin C given intravenously; say 1 gram or more. If you give a dose more like 250 mg., then the half-life is more like 60 minutes. If the dose is taken orally, as it would be by eating some fruit or in a tablet, then there is a longer time for the Vitamin C to get into the blood stream. In fact, the half-life continues to stretch out further and further until it is over 80 days as a subject gets close to scurvy. To your scientific mind, this sounds more and more like a complex system with a broad range of different relaxation times, from fast to long.

This means that the human bucket of Vitamin C is not draining to renal threshold nearly as quickly as the simple model would suggest. Let’s be even more specific. The level that the human human doctors have measured for the “fast bucket model” is Vitamin C concentration in blood plasma. Of course, different parts of the body absorb and store Vitamin C in different ways. Cells in the human immune system store Vitamin C as they absorb it from the blood. Likewise, collagen requires Vitamin C and it’s kept around for that too. In short, there are many tissues in the body that have their own uses for Vitamin C; and so there is a kind of balance between how much is circulating in the blood and how much various tissues in the body have kept. As the level of the vitamin in the blood gradually decreases, then these tissues will give their stores of the vitamin back up in a way that maximizes that amount of time it takes for the individual to become really sick.

### Intravenous versus oral dose: bioavailability

Also, you find that the only way to deliver a spike of Vitamin C like in your sketch above is with an intravenous injection; that is, straight into the blood stream. Unless you’re a human under a human human doctor’s care, that’s not very likely how you’ll take Vitamin C. You’ll swallow a tablet of some form or other instead. It takes much longer for the total amount of the Vitamin to enter your blood stream in that case. Here comes another factor that your doctor is aware of: “bioavailability”. Think about the area under one of these Vitamin C plasma level curves: that area would have units of Vitamin C concentration times time. In other words, something like micrograms over liters times seconds. A scientific person would write $\mu g L^{-1} s$. It would be a measure of how long the dose lasted in your blood stream.

Suppose you give a person a dose of 100 mg of Vitamin C intravenously and sample their blood for a while. You get a curve like that earlier sketch except it falls toward the “renal threshold”. Then, you give them the same dose in a tablet and take blood samples. You could then make two different graphs of how the plasma concentration changed over time and measure the areas under these two curves. You find that the human human doctors say that if the areas are exactly the same, then the vitamin was highly “bioavailable”. On the other hand, if the area under the intravenous-delivery curve was a lot greater than the area under the tablet curve, they say that the vitamin was not very bioavailable.

What human human doctors have found is that a high dose of Vitamin C in tablet form is typically less bioavailable than a smaller dose. In fact, the bioavailability also depends upon the form of the vitamin; say ascorbic acid as opposed to say, an ascorbate.

You begin a hunt for the human human doctors’ data.

### Some data

Your research takes you to the following: Vitamin C pharmacokinetics in healthy volunteers: Evidence for a recommended dietary allowance by Levine et al (Proc. Natl. Acad. Sci. USA, Vol. 93, pp. 3704-3709, April 1996). The doctors worked on seven healthy young human males. Here are four curves being shown: a lower dose given both in tablet form and intravenously, and then a higher dose also given both in tablet form and intravenously. [Click on the chart for a larger version.]

Effect of different Vitamin C doses

If you look at the “spikier” curves, you can judge for yourself how the half-life changes with dosage. For the 200 mg dose, the level begins at around “200 µM” and doesn’t fall to around 100 µM for about an hour. But for the 1250 mg dose, the level spikes to nearly “800 µM” and quickly falls to 400 µM in only 30 minutes. That appears to be that bucket draining, and even more quickly for higher doses, just like the human human doctors say.

You know from reading human science that “µM” stands for micromole. The “micro” part means the unit is one millionth of the whole. The “mole” part refers to a count of molecules: a really really large number of molecules (called Avogadro’s number). For Vitamin C (ascorbic acid), a mole turns out to have a mass of 176.12 grams. So a µM of ascorbic acid is 176.12 µg worth of the stuff. The next thing about those graphs is that the plasma density is not directly given. You have found that there are two typical reference volumes used in this kind of work: a liter and a tenth of a liter. Almost always, if you read in terms of moles then it’s referenced to a liter; and if you’re given units in grams then it’s referenced to a tenth of a liter (also called a deciliter or “dl” or “dL”). Since we’re reading micromoles here, the reference plasma volume is liters (written “l” or “L”).

Just for reference, the plasma level of Vitamin C that a human will be getting scurvy at is about 30 µM/L, which is also when they’ve only got a total of about 300 mg left within their body. Less than that and you’re getting quite sick. And at this level, they are losing Vitamin C only at around 2.5 mg per day. Here, you’re seeing a plasma level of about 70 µM/L at the baseline or the “renal threshold” level.

You want to consider more deeply  some of these numbers to keep things straight in your minds before you go any further. First, according to Wikipedia, total human blood volume is around 5 L in healthy males, who are the test subjects here. However, about 45% of that is circulating blood cells; the other 55% is plasma, the liquid part. That gives us around 2.75 L of plasma. This suggests that the total amount of Vitamin C at the maximum dosage level reported for Subject 3 in that graph (of 1250 mg) in his plasma is only about 800 µM/L x 2.75 L x 176 gm/M = 387 mg. That is a factor of around 300% less than the dosage administered. In other words, if Subject 3 had had a normal plasma volume of 2.75 L and a dose of 1250 mg (which is 7,100 µM) were delivered intravenously, you would expect a density of 2,580 µM/L instead of the 800 or so that show up.

Well, you read back and find that the 7 participants in the Levine study are all healthy young men. You go and find data on such individuals. For example, there is Erythrocyte, Plasma and Blood Volume of Healthy Young Men: Relationships to Body Size and Aerobic Fitness, by Sawka, et al. You look at this interesting graph from their study, showing a simple regression curve to predict, with reasonably accuracy, the plasma volume of healthy young human males:

Prediction of blood volumes

First, you note that “erythrocyte” refers to cells in the blood, such as red cells, etc. The sum of this and plasma should be the total blood volume. Suppose that instead of being the run-of-the-mill human male, Subject 3 were a large, aerobically fit specimen, with a lean body mass of over 100 kg (22o lbs). To put an intravenous dose of 1250 mg into his blood and come out with around only 800 µM/L instantly, he would have had to have had a plasma volume of 8.875 L. Looking at this graph, that seems highly implausible. Even a very healthy and fit NFL lineman or NBA guard might come in at, say, 6 L of plasma, accounting for size and fitness factors. (You like watching human sports in your spare time in orbit.) Imagining that their Subject 3 was such a big, fit specimen, and giving him 6 L of plasma, you get an instantaneous density of about 1,200 µM/L. And the 800 µM/L that is reported is about 66% of that. It is possible, considering the rapid rate of initial decrease of the vitamin, together with differences in the site where the vitamin was administered and the site where blood samples were taken, that this discrepancy is not unreasonable.

You repeat the calculations for the 200 mg (about 1.1 µM) intravenous dose and his response, and you see that the initial concentration is reported as around 200 µM/L. To get this reported Vitamin C density, Subject 3 would have to have about 5.7 L of plasma; and this is quite plausible. You assume then that the greater discrepancy at the higher dose can be attributed to the faster relaxation rate of the vitamin in that case.

Still, Subject 3 in the Levine study appears to be a fellow with a much larger than average plasma volume. At his baseline plasma concentration of about 70 µM/L, and giving him a 5.7L plasma volume, he’s got around 70 mg of Vitamin C in total in his plasma. But wait a second. You’ve already read that a human is about to get scurvy when their total body supply is around 300 mg or less. Here, they’ve just been dosing this big specimen with straight shots of Vitamin C into his veins and he seems to have far less than someone with scurvy.

Well naturally, you recall, the vitamin is being stored in more places inside him than just his blood. One place it could be going is right out of him through his kidneys. That’s what the human human doctor thinks is happening. Anything more than 70 µM/L in the blood is going to be swept out by the kidneys and give the expensive urine, according to their theory.

Another place it could be going is into various cells and tissues that store and use Vitamin C. In fact, you find that this is true. This is reported in a much larger study by Jackson, et al. Screening for Vitamin C in the Urine: Is it Clinically Significant? Jackson reports that “Vitamin C disappears first from the urine early in blood or tissue depletion. Plasma levels fall next and tissue levels (such as leukocytes and platelets) are the last to fall. In healthy human tissues the maximum vitamin C pool varies. Data from the literature gives three different ranges, from 1500 to 5,000 mg (20, 22, 32 mg per kg of body weight).” Based on this information, that Subject 3 might have had as much as 3000 to 4000 mg of Vitamin C total in all his tissues; and that is certainly much more than was being flushed out of his blood supply through his kidneys. That’s assuming that Subject 3 has a body mass of 110-130 kg and that the 32 mg/kg estimate is correct.

Jackson observes, quite humorously you think, that goats are among the animals that produce their own Vitamin C, unlike the poor humans. A 70 kg goat will produce over 2 grams of Vitamin C per day, which is of the order of so-called “mega-doses” for humans. Under stress or disease conditions, that goat will produce, on its own, as much as 13 grams of Vitamin C. Then you find that some human human doctors are doing exactly this sort of treatment, using intravenous injections of Vitamin C in cancer patients and with good results.

You read on. Jackson reports on a much larger base of measurements in some thousands of individuals who had Vitamin C levels sampled both in their plasma and in their urine, in order to establish a level at which the kidneys would kick in and flush out any “excess” Vitamin C. Here is their table of results:

Vitamin C in plasma & urine

Unfortunately, the table as published uses the alternative units for Vitamin C concentration; namely, mg/dL, as opposed to the µM/L in the Levine study.  Just to make this a little easier to put into the context you’ve been working in, you put this information into the form of a graph showing the percentage of the 6,537 subjects having a given Vitamin C concentration in their plasma in the same units that Levine was using.

Histogram of Vitamin C plasma levels

This distribution has a mean value of 74 µM/L and a standard deviation of 42 µM/L. Levine’s Subject 3 would be right at average with his baseline level. However, you see that there are many subjects in the Jackson data set with plasma levels of Vitamin C much higher than average. In fact, it goes without saying that if the average is 74 µM/L, then half of the subjects have values higher than that. Something around 5-6% of subjects have plasma values over twice the “renal threshold” in Levine’s Subject 3. You’ve seen probability distributions like this before; the human scientists call them heavy-tailed distributions. You think that it is very natural that a system that had a broad range of relaxation times would show a heavy-tailed distribution like this; it makes perfect sense to you.

Jackson considers the rate of Vitamin C loss in urine in his work. Data available on the tubes of the Internet state that the average volume of urine excreted by the average human is from 1 to 2 L per day. Also, your average human “goes” 4 times a day. If one was excreting, say, 1250 mg of Vitamin C within the first few hours of ingesting it, and if their total urine volume was 0.5 L (5 dL), then you’d expect to see a density of 250 mg/dL. Even if you account for a bioavailability of only 1/3rd of that (the rest passing straight through the gut, you should see values of over 80 mg/dL in Jackson’s data. Still Jackson does not appear to be reporting such high levels, although there is certainly a trend in his data for higher levels of Vitamin C in urine to correlate with higher plasma concentrations.

Levine also looked at the excretion of Vitamin C in urine. His results seem much more conclusive to you than Jackson’s. Here is a graph from his paper.

Excretion of Vitamin C in urine

You can see that his findings seem to show that all of the Vitamin C in high intravenous doses is excreted in urine. The tell-tale feature in this graph is “Insert B”. That little chart shows the relative amount of the vitamin that winds up in the urine as a function of the dosage. For anything above 500 mg, all of the vitamin seems to be lost in urine. At around 200 mg or so, only half is lost in the urine.

But hang tough. Doesn’t this seem odd, just a little. At a 200 mg/day dose, that means the body is pealing off 100 mg for use and eliminating 100 mg. But at 500 mg/day, all of it is eliminated and none is consumed. This seems to be a rather odd economy. One might expect to see that demand would increase as supply becomes “cheap”. For example, at high availability of Vitamin C as a resource, then consumption would be inexpensive. As supply goes down, then only the most critical consumers of the Vitamin C resource would get an allocation. Here, you are observing, apparently, that at a high level of supply of this resource, making it virtually free, no one is buying. As it becomes more scarce, buyers show up. The mystery of Vitamin C continues.

You go back to some more data from the Levine study.

Evolution of Plasma Levels

The top curve shows explicitly the evolution in the plasma levels of each of the 7 subjects in the study. All seven subjects appear (at least to your eyes) to be showing a steady increase in plasma levels all the way out to the end of the observation point of the trial. You suppose that it is anyone’s guess what might have happened if these individuals had continued to receive, say 2,000 mg of Vitamin C per day over several years (like Jackson’s 70 kg goat). Nonetheless, Levine concludes that there is a plateau level achieved in his 7 subjects and he shows these plateau levels versus daily dose in the bottom chart. You can see that the plateau level achieved using the current RDA is about 1/3rd of what is achieved at a daily dose of over 1,000 mg. You wonder about the value of the current RDA.

This brings to your mind another study you’ve come across. This is Hickey, et al on the Pharmacokinetics of Oral Vitamin C. This study was done in England and compared standard tablets of Vitamin C to liposomal Vitamin C in terms of their bioavailability. You browse once more the paper in its entirety. Hickey demonstrates plasma levels of Vitamin C as high as 400 µM/L using oral doses of liposomal Vitamin C; and plasma levels of over 200 µM/L are easily achieved. In effect, Hickey employed a sort of time-released capsule that delivered and maintained high plasma concentrations without having to go to an IV drip. The aim was treatment of cancer patients, again. Apparently, this “renal threshold” level is not cast in stone (not even kidney stone.)

You go back to Levine. He and his fellow researchers also investigated how various daily doses of Vitamin C impacted the amount of the vitamin present within certain cells of the immune system. You appreciate the attention to detail in this study. Here’s one of their graphs showing what they found.

Vitamin C uptake in the immune system

This graph shows the amount of Vitamin C within three different kinds of human immune cells; namely, neutrophils, monocytes, and lymphocytes. These are the human body’s first responders to trouble. They store and use Vitamin C to do their work. You can see from this graph that these seven young men absorbed Vitamin C into their immune cells until they got about as full as they could be at something like a daily dose of about 200 mg per day. The units are now in “mM” or millimoles (1/1000 of a mole). For Vitamin C, 1 mM = 176.12 mg. So, you’re seeing here that the neutrophils have absorbed and stored around 200 mg, the monocytes have about 525 mg, and the lymphocytes have about 600 mg, give or take. That’s around 1,300 mg of Vitamin C circulating within immune cells. You seem to see here that these little warriors have all of their ammo belts filled up and they are ready for battle if they are getting more than 200 mg /day.

What you don’t seem to be getting here though is these little fighting men’s rate of use of Vitamin C when they do go out to the battle front. They start off at around 0.5 mM for the neutrophils and about 1.5 mM each for the monocytes and the lymphocytes. You see that run up as Vitamin C is made more available to them. But you don’t see what happens during a fight. Apparently, Levine’s seven subjects were depleted at the start of the study. They began with around 2 mM total (350 mg) and went up to just under 8 mM (1,300 mg) as dosage was increased. That’s a factor of nearly 400% increase in stored Vitamin C in the immune system cells that were observed here.

Of course, as Levine’s study clearly states, you are looking at healthy young human males here. Jackson has told you that total body stores of Vitamin C can be as high as 5,000 mg; and here you see around 1,300 mg in three kinds of immune cells as they are healthy and prepared to fight. That leaves as much as 3,700 mg floating around somewhere else.

There is perhaps one other aspect of this amount of Vitamin C storage that is not precisely accounted for. It’s simply that the data presented are for 7 different subjects and yet there does not seem to be much variability in the amount of Vitamin C stored in these cells. If a single neutrophil has such and so capacity for Vitamin C, then the total storage of the vitamin should depend on the number of neutrophil cells. Individual humans of different sizes ought to have different total blood volumes, and hence, different numbers of neutrophils. One might expect more variation in total Vitamin C storage than we are observing here. That would be consistent with Jackson’s information about total tissue Vitamin C correlating with total body mass with a multiplier (his 20, 22 or 32 mg/kg).

# Let’s take a breather

What have you learned so far?

1. Vitamin C, orally or intravenously, goes into the blood stream first.
2. It is assumed by many researchers that plasma levels above some (renal) threshold (around 70-80 µM/L) fall rapidly (exponentially) to this level. The rate of loss of Vitamin C appears to vary significantly, from a half-life of 30 minutes after a large intravenous dose to over 80 days at the onset of scurvy.
3. The current RDA (90 mg/day for men) does not seem to achieve immune cell saturation and some scientists (Levine et al) have argued for the RDA to be raised (to 200 mg/day for men) to achieve saturation of immune system cells. Depletion of Vitamin C from the immune system under conditions of stress or disease was not investigated.
4. Other mammals of a size similar to humans (goats) that can produce their own Vitamin C create daily doses of 10 times the 200 mg/day recommendation and even more when under stress or diseased. Other primates of a similar size to humans (gorillas) get around the same amount (4 grams or so) through their diets.
5. Oral doses of liposomal Vitamin C achieve plasma levels as high as 200-400 µM/L.
6. The total amount of Vitamin C in plasma can be relatively low in comparison to the amount stored in circulating cells of the immune systems. That is, of Vitamin C within the whole blood, only 50 mg might be dissolved in the plasma while over 1,000 mg may be contained within immune cells.
7. Estimates of total body stores of Vitamin C in healthy individuals vary (20, 22, or 32 mg/kg of body mass).
8. Observations of the rate of excretion of Vitamin C in urine seem to yield conflicting or inexplicable results; e.g., less consumption at higher availability.

What you should be seeing is that, as your teacher back on your home planet used to say, “the goes-inna should equal the goes-outa”. If you have a human body, and you are putting in, say, 1000 mg/day of Vitamin C, then you ought to see a total of 1,000 mg coming out in urine or stool or sweat or breath or whatever, plus what might go into body stores, plus what is consumed by various bodily functions (and turned into some other metabolic by-products).

Each of the research papers that you’ve looked at attacks just a small part of this global picture. Levine shows you some features about the absorption and storage of Vitamin C in three kinds of immune cells (and there are more), about urine excretion, implicitly about stool excretion (through bioavailability estimates), about blood levels. However, he doesn’t address immune system consumption in cases of chronic or acute disease or stress. He also doesn’t seem to address the odd phenomenon of different half-lives at different dose levels or different excretion versus consumption patterns at different dose levels (higher doses yield less consumption). His data are based on seven healthy young men. You’d like to know more biometric information about his subjects: height, weight, plasma volume, total blood volume, lean body mass, and so on.

Hickey tells you something about the bioavailability of Vitamin C in oral doses of a liposomal format and provides data that conflicts with Levine on achievable plasma levels of the vitamin. His data are based on only two subjects. Jackson gives you information about plasma levels and how they correlate with urine excretion for over 6,000 subjects. He gives you a simple calculation about the depletion rates of Vitamin C based on total body stores, running from 5,000 mg down to the onset of scurvy at 300 mg. The accuracy of his strip test for Vitamin C in urine may be questionable. His histogram for Vitamin C in plasma levels shows a non-Gaussian, heavy-tailed upper distribution that he does not account for. There is a deep mathematics in this, as some humans have been discovering.

Still you seem to come away with more questions than answers and no really good sense of what an optimal dose of Vitamin C for humans might be. You don’t know what the total consumption of Vitamin C within various body functions might look like, and how that consumption would vary with different conditions. For example, you know that Vitamin C plays a role in collagen formation; what would collagen consumption of Vitamin C look like in an individual in a post-operative condition and rebuilding collagen? Vitamin C plays a role in the human immune system. What would consumption of the vitamin look like in an individual with the common cold? with various forms of cancer? with adult onset diabetes?

## Vitamin C consumers

Vitamin C plays a role in several body functions as well as within a number of immune cells. You list these out:

1. Formation of collagen, including scar tissue, blood vessels, and cartilage.
2. Formation of carnitine, critical for delivering fatty acids into mitochondria for energy production.
3. Formation of the neurotransmitter norepinephrine from dopamine, important for concentration and muscle tone.
4. Stabilizing peptide hormones, which include growth hormone, insulin, and leptin (which regulate overall metabolism and fat storage).
5. Modulation of tyrosine metabolism, an amino acid used in protein production, including CoQ10 and various neurotransmitters.
6. Metabolism of microsome, essentially waste left over from dead cells.
7. Preventing the oxidation of iron and copper.
8. In the immune system, roles in phagocytes, cytokines, neutrophils, lymphocytes, and monocytes.
9. Antioxidant to control oxidative stress.

That seems to be a big and important list to you. Levine’s study has given you some information about the Vitamin C stores for #8 on the list. You might conclude that if some part of #8 can store around 1,300 mg or so and from Jackson you have estimates of total body stores of the vitamin up to 4,000 to 5,000 mg (depending on body size), then the rest of this list would seem to involve storage of perhaps another 2,000-3,000 mg. Clearly, these nine different functions for Vitamin C may involve both storage and consumption; and consumption would depend upon the rate of use under different conditions. For example, rebuild cartilage or damaged blood vessels and consumption in #1 increases. Go on an exercise campaign and consumption for #2 will increase. Controlling body weight in an individual with metabolic syndrome or adult onset diabetes may drive up consumption in #4. Dealing with an infectious disease or cancer will drive up use in #8. Smoking, higher levels of exercise, and various degenerative conditions will increase consumption in #9. Levine counsels you that, for healthy young male humans at least, then you don’t have to fret too much about any of this stuff; and 200 mg/day will do them nicely.

But what if the subjects are stressed-out, slightly over-weight, not so healthy 50-year-old humans? What if they just can’t seem to shake off that flu bug that their kids keep bringing back from school? What if they are run-down, unable to concentrate, and always yelling at their spouse? What then? If Levine saw his seven healthy young subjects add around 1,000 mg of Vitamin C to their immune cells during his study, maybe these not-so-healthy individual humans could do with a real boost of the vitamin too.

What if?

# Back to the buckets

You think more about that deep mathematics. You decide to redraw that graph you’d done of Jackson’s 6,000+ subjects’ Vitamin C blood plasma levels on log-log scales. This is what you find:

Log-Log distribution of Vitamin C plasma levels

This absolutely looks like a power law, except perhaps for the first point. If that had been “levered up” to, say, 80% or so, then it would be a classic power law distribution. That pesky point shows that only 12% or so of subjects had very low blood plasma levels of Vitamin C, at around 17 µM/L, which is technically scurvy. These levels were tested through a laboratory in Kansas, USA. The food supply in the USA has all sorts of sources of Vitamin C. Hell, it’s added as a preservative in many highly processed foods. Perhaps the natural expectation for a population of homo sapiens without such an industrialized food supply would be for a much larger fraction to show these very low blood plasma levels. What you’re seeing here is a side-effect of the humans having figured out something about the vitamin. If only you’d arrived 200 years earlier. You bet that that first point would have been much higher back then, before the humans had the industrial chemistry to put volumes of Vitamin C into their food.

The steepest part of this curve has a power of over -5, which is huge. You can see, though, that at the upper end of the chart, that steep drop seems to be leveling off: the power has dropped to just over -2. You wonder if Jackson’s data are questionable, although it was a large study, over 6000 subjects. That roll-off at the upper end may also be due to the humans’ informed behavior; that is, you expect that the subjects with the highest plasma levels of Vitamin C are purposely taking high-dose supplements. In general, the use of supplemented food and just plain supplements will tend to drive measured plasma levels upwards.

# What is Vitamin C’s mission?

But you know that whenever you see a power-law curve like this, there are at least two underlying exponential curves floating around behind the scenes somewhere. And yet, something is missing in the logic. You look again at the list of functions of Vitamin C and the curves for plasma densities, and it occurs to you to ask, “What is the mission of Vitamin C?” Is Vitamin C’s mission to circulate in the blood? Or is Vitamin C’s mission to support the functions on that list, none of which directly involve being in plasma?

Of course, Vitamin C’s mission is not just to float around in the plasma; that is just a vehicle for getting it to where it is needed. That would be like saying a Marine’s mission is to ride on a ship. One would not measure the effectiveness of Marines on the basis of how many were on board naval vessels; you would look at how they performed once they arrived at their destinations.

That list of functions of Vitamin C involves construction, repair and protection of tissues. Any organism is a complex, and highly redundant collection of microscopic, cellular elements. Unlike the machines and computers that the humans have learned to make, the individual components of an organism do not come into being in a state of perfection; and they are very frequently damaged in operation and must be destroyed and replaced. This is an on-going process. In a machine, you might find one or two redundant components for any function, say, a power supply or a CPU. Each of these was tested in the factory before being installed in the machine. The overall machine cannot stand too much failure in any single component before the entire system fails. In contrast, an organism is continuously accumulating damage at a cellular level, and actively working to defend against that damage and then to repair that damage by removing dead cells and introducing replacements.

In short, you think, the mission of Vitamin C is to extend life, and perhaps, to increase the quality of life. How much is in the plasma at any given time is not a measure of its effectiveness. It occurs to you that the relaxation model is part of the physics of inanimate objects, but that the context is different, it should be a science of living things.

You look back at Levine’s data and you can see that whatever might be the story about how much is in the plasma at any instant, the amount in the immune cells was pumped up by 3 or 4 times during the interval that the subjects took higher doses. Plainly, the higher plasma levels have lead to an increase in Vitamin C storage in the immune system, which is part of Vitamin C’s mission to extend life.

You think back about the amount of Vitamin C made on a daily basis by those mammals that can produce it, and about the amount of the vitamin consumed in diet on a daily basis by those primates living in the wild that cannot produce it. For mammals and primates of a similar size as a human, say 70 kg, they are making or consuming about 4 grams per day. This is around the total body story of Vitamin C in a very healthy individual, of whatever mammalian species: goat, gorilla, or human. That means that if, for any reason at all, their bodies lost the entire store of the vitamin for some reason, it could be maintained within a 24 hour period.

## Human life-span

So you are lead to consider human life span, and more broadly, the life span of other species of organisms as well. It seems that human life spans have been studied for some time and that an exponential law has been discovered that describes the process fairly accurately, the Gompertz-Makeham law.  You look at this law for human life span, or mortality, and you think, hmm, this is just a survival function, a hazard rate, a part of reliability theory about how things break down and fail to complete their mission.

You know that if the hazard rate is a constant, then a population dies off at an exponential rate. Actually, this is just like what the human human doctors believe about molecules of Vitamin C at high levels in the plasma: their survival rate is a decaying exponential because the kidneys filter them away at a fixed rate. Well, it isn’t such a fixed rate at all: the rate varies depending on the density, stretching out the time that the molecules of Vitamin C can survive because really the kidneys begin to put some back once the level gets lower. That’s just the way that kidneys work.

In this little block, I’m going to break with the main text’s story line about visiting alien scientists. Earlier, I gave you a simple mathematical model for a relaxation system, an exponential decay; the sort of thing that a scientifically-minded medical researcher might apply to the removal of Vitamin C from plasma, as we’ve observed. We’ve also seen that researchers have found a wide variation in the relaxation rates (or half-lives) depending upon the body’s existing stores of the vitamin. We could get a similar effect with a simple bucket: we could just squeeze the spigot valve shut as the volume of water left in the tank decreased. That might do the trick. However, that doesn’t seem to fit the way in which Vitamin C is stored or depleted from the body. It’s as if there were a variety of internal storage “buckets” and each one is running out at a different rate. That’s more like filling up a collection of buckets of different sizes with different spigots. Then we’d watch the entire volume of water in all the buckets flow out. Maybe we could work out some way of sizing the buckets; for example, the second one could be half the size of the first, the third would be half the size of the second, and so on down the line. That would be a kind of “power law”; the power would be 2. We could play around with our set of buckets by choosing different powers.

It turns out that exactly this kind of situation occurs in the physical sciences all the time. I studied this sort of situation for my Master’s degree in electrical engineering. I was looking at the dielectric properties of crude oils. Crude oil is nasty stuff. It comes up out of the ground with all kinds of dirt, water bubbles of all sizes with dissolved salts, and a broad range of different hydrocarbons with different electrical properties.

Let’s say I have some molecule in that oil that has a positive charge on one end and a negative charge on the other. That’s called a dipole (because it has two poles). I apply a strong electrical field. Then that dipole will align itself with the field, just like a compass needle to magnetic north and south. The way that works is exactly like one of these simple relaxation phenomena, like emptying a bucket of water.

But what if I have some goop like crude oil with hundreds or thousands or millions of different kinds of dipoles, each with its own relaxation time. If you measure the way in which such a complex system relaxes, you’ll find a much more spread out result. The fast dipoles relax quickly and the longer dipoles seem to stretch the process out indefinitely. The overall relaxation depends on how much of each kind of dipole there is. Imagine that you have such an electrical bucket filled up with charge. You let it begin to discharge and measure how the total voltage decays away.

## Stretched Exponentials

In fact, exactly this kind of goopy system was first understood by a German scientist named Rudolph Kohlrausch as early as the mid-19th century. Kohlrausch had just such a mixed up electrical goop in something called a Leyden jar. He’d get a static charge on this goop and then watch how the charge relaxed away with time. His solution to the odd mathematical curve that he discovered is now called a stretched exponential or a Kohlrausch function (in his honor). A stretched exponential looks like this

$f_\beta (t) = e^{ -t^\beta }$

I’m not going to assume that all you readers grok this pretty little piece of mathematics. The $\beta$ in this equation is a power that, numerically speaking, is usually between 0 and 1. One way to think about this would be that throwing in that power of $\beta$ is rather like controlling how the spigot on the bucket gets closed down as there is less water inside. I’ll leave you, if you are interested, to explore this model further.

I do think that I should provide you all with a quote from this paper, however, just to give you an idea of how broadly the stretched exponential has proven of value in science.

There is no doubt about its physical significance, they are encountered for example, in recurrence time statistics [1], long-term correlations on rare events [2, 3], fractal rain distributions [4], Earth’s magnetic field fluctuations [5], distribution of shortest paths in percolation [6], catalytic activity of molecules [7], distribution of potential energy in granular avalanche [8], trap times distributions [9], random walks on fractals [10], enstrophy flux in two dimensional turbulence [11], velocity distributions in sedimentation [12], sleep state transitions in humans [13], activation energy barriers [14], noncolloidal suspensions [15], dynamical heterogeneities [16], distributions of relaxation times [17], time step distributions in a protein folding model [18], distributions of film earnings [19]. Also, they are relevant in the study of the tail of distributions, for example, the study of velocity fluctuations [20], in velocity fluctuations in granular flows [21], velocity distributions in inelastic hard sphere systems [22], velocity increments in turbulent flows [23], turbulent scaling [24], velocity increments in weakly turbulence flows [25], quantum diffusion [26].

Wow, that’s a lot of applications. In fact, you can try this one for yourselves: Google “stretched exponential vitamin”. You’ll come back with all sorts of research papers on how certain vitamins respond dielectrically with stretched exponential relaxations. So it’s not that research scientists looking at vitamins are unaware of this piece of science; but they are not applying the model to how vitamins are removed from the body.

## Fractal Relaxation

Bet you didn’t think you’d see those two words together when you started reading this! Well, I really expect to lose a lot of readers at this point; but I’ll attempt to be as absolutely simple as possible. You may or may not have heard the word, “fractal” before this. I’ll explain the idea this way. It has to do with a pattern that has a kind of symmetry. If I look at a pattern of tiles on the floor, it might look the same if I shift over a few tiles. That’s called translational symmetry: I shift my viewing position a few steps and everything looks the same. Perhaps another pattern has rotational symmetry: that means that if I turn the pattern through some angle, it looks the same again. Then there’s mirror symmetry: if I look at my right hand in the mirror, it looks like a left hand. And so on. Fractals have scaling symmetry. That means that, for example, if I magnify a pattern by, say, 10 times then it looks the same as it does when magnified by 100 times or 1000 times. Or perhaps to get the pattern to look exactly the same, the power has to be 1.7 or 3.995 or some other number.

Sometimes the pattern looks exactly the same, like what happens with lots of tile patterns that show up on floors or walls or ceilings. Sometime, in the case of fractals, the pattern is only statistically similar; that is, a probability distribution looks the same as we scale up or down in looking.

What fractal relaxation is about is something like what I meant in describing a group of buckets that scaled in size in some way. If you set up a series of buckets that all went down in size by ½ at each step, then you could go 3 steps sideways and magnify the buckets by a factor of 8 and everything would look the same. That would be fractal symmetry.

In the interests of keeping what I’m writing here at least somewhat digestible by a reader of good intelligence (I assume you’re that if you’ve made it this far), I am going to be hand-wavy instead of rigorous here. For those who care to explore this in more depth, you can start with the Wikipedia article on fractional calculus. Using the tools of the fractional (or fractal) calculus, one writes down a fractal relaxation differential equation and then solves it using a fractional integral operator. [Sorry, that’s what you do though. I said this was about 21st century mathematics.]

I’m not even going to mess with your mind on what the solution looks like mathematically. It’s pretty bad. Much worse than the stretched exponential. [It has gamma functions in an infinite series! Hahahaha! Imagine a mad scientist laugh at this point.] Instead, let me just show you examples of the three sorts of relaxation models that I’ve covered so far: the simple exponential, the stretched exponential, and the fractal relaxation models.

Three Relaxations

There are a couple of parameters to play with in the fractal and stretched exponential curves. For this simple compare and contrast, I set the time constants to 1 for all three and chose the scaling exponent (aka the power) to be ½.  You can see that the simple exponential that we started with falls off even more slowly than the other two curves, but it goes straight down to 0 very quickly. The fractal relaxation falls off most quickly and then hangs on at a level of just under 20% of maximum. Similarly, the stretched exponential falls off faster than the simple exponential, but seems to hang on at around 20% of maximum as well.

Another way to have shown these curves would have been to adjust the parameters so that the fractal and stretched exponential showed the same half-life as the exponential; then they’re very long “tails” would have stretched out even more than what is seen above.

What the current medical thinking about high vitamin doses amounts to is, mathematically, an exponential relaxation plus a baseline below which a completely different set of mechanisms are supposed to take over. In contrast, employing either the fractal or stretched exponential models would imply is that there is one, integrated model for the relaxation of stored vitamin towards complete deficiency. These models would also begin with the notion that the human body is a complex, yet integrated and orderly, collection of subcomponents, each with its own relaxation to deficiency; and yet the whole system relaxes in a collective manner.

In contrast, the literature on the application of fractal relaxation (often also called a Mittag-Leffler function) and the stretched exponential, recognize that these functions are usually a transient phenomenon that smoothly proceeds into a power law. These appears to be exactly consistent with Levine’s data on his Subject 3; in other words, the initial response to the intravenous injection has the form of a stretched exponential followed by a transition into a slowly decaying power law function. By performing a least-squares curve fit to Levine’s data for Subject 3 for the 1250 mg injection and including a data point for “scurvy” (assuming 0 plasma Vitamin C status at 2000 hours), I have come up with the following curve.

A model for Vitamin C absorption

As you can see, the fit is quite good. Better yet, it involves only a single, integrated view of the relaxation of Vitamin C status from good health out to deficiency disease.

So human life span begins to look a lot like a problem in reliability and Vitamin C begins to look like a factor that could impact reliability and, hence, longevity. For the Gompertz-Makeham law, there are two components to the hazard rate, a constant term and an exponential term. The constant term has nothing to do with the age of the individual; it would go to constant risks from outside (extrinsic causes), like accidental death. The exponential term, that becomes more important as age increases, would go to increased risks from internal problems (intrinsic causes) like degenerative disease. Something like Vitamin C can’t help much for an individual human that falls over a cliff and is mortally injured. However, Vitamin C’s mission is precisely tuned to deal with those intrinsic issues of degenerative disease.

## Aging and Vitamin C

It begins to look like human aging is a problem involving an accumulation of random damage to various, otherwise redundant, internal components; that is, tissues. A human who smokes tobacco will gradually accumulate damage to lung tissues. A human who has elevated levels of homocysteine will gradually accumulate damage to arterial smooth muscle. Eventually, enough of the otherwise huge volumes of tissue in the lungs or arterial walls (in terms of total cell count) is damaged that a catastrophic failure occurs; and the individual fails.

The presence of Vitamin C, and other micronutrients, is useful in preventing and repairing this sort of accumulated random damage. In an organism, like a homo sapiens individual, there is a huge degree of redundancy at the cellular level; that is, every major organ is comprised of a significant number of individual cells. The death of any given cell does not signal great damage to the organ; in fact, this process of cell death is continuous. In a typical adult human individual, 50-70 billion cells die each day and are replaced. This is out of some 100,000 billion cells total in the human body. Once more, you see a survival rate model; this time it is for human cells and the rate is around 0.9994 per day. The hazard rate is just the opposite; that is, 0.0006 per day, meaning that that fraction of human cells die a normal death each day. These numbers go to the incredible redundancy in an organism like a homo sapiens individual: no machine could lose that many components every day and continue to survive for long.

Depending on the cell type, Vitamin C would be involved in the removal of the remains of the dead cell (microsome, above) and the production of components of its replacement cell.

In the Gompertz-Makeham law, the hazard rate has these two parts: a constant (for uniform external causes of mortality like falling off a cliff) and a variable and an increasing term (for internal causes like the accumulated degeneration of tissue that cannot be repaired). You write this down as follows:

$\mu ( t ) = R + A exp ( \alpha t )$

It ought to be that good Vitamin C status (and other micronutrient status) in an individual human would effectively reduce the accumulated and unrepairable damage to otherwise redundant tissues; that is, it should modify the $A$ and $\alpha$ terms in that equation. You know that there are two competing processes in human metabolism that manage the functions of cell death and replacement. There is a catabolic process that breaks down tissue and an anabolic process that builds it up.

On a daily cycle, primitive homo sapiens in a Paleolithic hunter gatherer state, as you’d found them 20,000 years ago, would exert themselves to obtain food. During this interval of exertion to hunt or gather, they are not eating much or at all. Then, they pause and rest, during which time they would eat and then sleep. As they exert themselves, oxidative stress levels are increased and damage is sustained, especially to the weakest cells. [If you want more information on this cycle in exercise, check this out on this web site.]

Rather than acting to protect the weakest cells in the organism, lowered levels of antioxidants during exertion would promote cell death in them. this would cull the weakest and most vulnerable cells, in a kind of “survival of the fittest” manner. Later in the day, after eating and during rest, the elevated levels of Vitamin C and other micronutrients would then play a role in the processes of removing dead cells and building their replacements.

## Dynamic Kinetic Stability

There is a kind of stability that living creatures have (carbon-based or silicon-based or whatever). A perfect image of this is a hummingbird floating in front of a flower. In one sense, it is balanced and stable. In another sense, there is so much going on dynamically to maintain that stability. It is not the stability of a marble that has come to rest in a bowl (low potential energy). It is not the stability of a glass of once-hot water that has cooled to room temperature (thermal equilibrium). It is a dynamic kinetic stability (DKS) that is the hallmark of life. Only while it’s alive can the hummingbird achieve that floating balance. If it dies, then it will come to rest at the lowest place around it and its temperature will come to that of its surroundings. In that case, the relaxation model takes over, and DKS ceases.

Purple-throated carib hummingbird feeding

The same DKS model applies to a human standing on two feet. He appears stable, but his balance is a trick of continuous fluctuations of a significant proportion of all of his muscles. Only while he is alive can he maintain his stance.

Here is another example: the floating balance of Vitamin C in the tissues of that human. It seems stable, and yet, it requires so much dynamic and kinetic effort, like that hummingbird in the air. It is the dynamic kinetic stability of life in action. Only while the human is alive can he maintain that balance of Vitamin C in his tissues. Like the beating wings of the hummingbird, there is a daily cycle of exertion & tissue stress followed by eating & tissue rebuilding.

The optimal dose will have to have something to do with this DKS cycle.

### Recent human evolution

You also recognize that modern homo sapiens is different in many ways from the ancestors that you studied 200,000 years ago. Back then, those creatures were deep in the process of becoming evolved hunters. This could imply a broader ability to obtain micronutrients, like Vitamin C, from animal sources than for their forest-dwelling primate cousins. In terms of modifications to digestion, that implies not only genetic adaptations like being able to digest lactose into adulthood, but also changes to the flora and fauna of the gut. Modern humans are comprised of many internal adaptive systems; not just the genome, but also the adaptive immune system and the adaptive digestive tract. Just as an infant will acquire some adaptive immunity from its mother, it will also acquire some beginning probiotic status too. Each of these initial conditions evolve within the individual in unique ways based upon life history. Acquired immunity may be similar in a given population of humans, since survivors would likely have had a similar history of contact with contagious diseases. Likewise, acquired gut probiotics would be similar based upon similar dietary patterns.

This means that human evolution, with respect to dietary adaptations, comprises more than just any genetic differences between modern humans and their primate ancestors. You also have to consider very local and very recent adaptations to the flora and fauna of the homo sapiens gut.

# Finally, sense emerges

Then, “Eureka!” as the Greek humans used to say. Suddenly the cycle of higher and lower concentrations of Vitamin C in the plasma after a meal begins to make sense to you. You can now see why such a cycle would have evolved. There is this daily cycle of exertion and rest, catabolism and anabolism, tearing down and building back up. Exertion captures the meal. Then the meal rebuilds what was destroyed in exertion. Higher Vitamin C in the plasma synchronizes with anabolism (rest and rebuilding). Lower Vitamin C in the plasma synchronizes with catabolism (exertion and stress). This is a daily cycle of dynamic kinetic stability.

You also check back and note that the production of Vitamin C from glucose kicks off a molecule of hydrogen peroxide ($H_ 2 O_ 2$), which is a significant source of oxidative stress. Hence, an organism that obtained all their Vitamin C from diet instead of from internal processes would transfer the oxidative stress burden of making a molecule of Vitamin C to the plants that produced it. There turns out to be an advantage after all in not making your own Vitamin C, if you can get all you need from diet. It is odd that the production of Vitamin C, a notable antioxidant, is itself a source of oxidative stress. Perhaps there is a balance in this reaction: one molecule is anti-oxidant and the other is pro-oxidant. This makes a ton of sense.

The mystery of Vitamin C is becoming less mysterious to you. And there is more in the same vein. It turns out that Vitamin C status is a controlling factor for Hypoxia-Inducible Factor 1-alpha ($HIF1 \alpha$). Low Vitamin C status, among other things, up regulates $HIF1 \alpha$ which, for its part, then expresses hundreds of stress related genes. This role of Vitamin C is an aspect of a longer cycle in original human life-styles; namely, between winter and summer. In summer, there is that daily cycle of exertion and stress together with eating, rest and recovery. In winter, when food is scarce and the individual must survive off of stored reserves, including those of stored micronutrients like Vitamin C, then it is important to conserve stores of the vitamin to avoid scurvy. Lowered Vitamin C status in the plasma through the long times of near starvation in a winter or famine period, then, turns on a system of genes that manage stress by retaining and utilising Vitamin C elsewhere.

Here is another advantage to not producing Vitamin C, in effect, that vitamin’s status can be used as a proxy for lowered nutritional intake, thereby switching on a cascade of genes that manage stress due to that condition. In a mammal that produces its own Vitamin C, the high natural levels would always mask poor nutritional status. Not so in humans: poor Vitamin C status turns on a set of stress genes: the winter genes, the starvation genes, the long-sea journey genes. What then for poor Vitamin C status in a human that is otherwise getting high levels of macronutrients? That is, what about the typical undernourished and overfed North American homo sapiens? They could have up regulated $HIF1 \alpha$ and expressed a broad range of stress genes. They’ve put their bodies into a totally artificial pro-inflammatory stress-mode that would push energy storage (i.e., fat build up) in an unnecessary manner. That sucks, you think. Poor them.

## What about an optimal dose?

You think you are, at last, close to figuring out optimal doses for Vitamin C. If one is a very healthy young (very nearly at the cusp of the Gompertz-Makeham curve at about age 20) male human, then perhaps Levine was dead on and somewhere around 200 mg per day is not too bad a choice. This is close to the minimal mortality point for humans, it seems.

One can tell fairly easily if one is young. But how do you tell if you are healthy? The Gavrilov & Gavrilova studies suggest that many, if not most humans have already accumulated a high level of damage to redundant tissues by the time they are infants. The older one gets, the more critical it becomes to deal with this accumulated damage, as indicated by the Gompertz-Makeham curve: an exponential increase in mortality after the age of 20 in humans! If you were a human, (and thank the stars that you are not!) you’d be getting all the damn micronutrients that you could, including Vitamin C. You thank your evolution that you are a silicon life form and none of this stuff applies to you. You look at actuarial tables for humans living in the United States of America. It’s pretty easy to fit a curve to this data set for humans between the ages of 30 and 90. Here’s what you get:

Gompertz-Makeham for USA males

The formula is

$1 - e^{-\lambda t - \frac {\alpha} {\beta}\left (e^{\beta t} - 1 \right)}$

where $\alpha = 0.000000942132$, $\beta = 0.11055$, and $\lambda = 0.000061405$. Of course, that isn’t the entire “life table”, just the section for ages 30 and above. The entire life table looks like this:

2010 Life Table, USA Males

Apparently, USA human males between the ages of 16 and 25 or so are prone to accidental death. The exponential aging part appears to take over for ages above around 10 years. As well, above 100 years, the exponential mechanism appears to slow down. Mortality falls sharply from the moment of birth until age 10, when it begins to climb rapidly again. As the Gavrilov & Gavrilova studies suggest, perhaps many children succumb to early damage in the years between 0 and 10. Those who survive begin to fail at an exponentially increasing rate, at least in part due to accumulated stress on those tissues that were exposed to the greatest early damage. Above age 30 years, this exponential degenerative term becomes dominant.

At between 2 and 4 gm/day, other mammals of equivalent size to humans are making or consuming their entire body’s store of Vitamin C, or more, each day. This seems extreme in the absence of a clear and present indication of illness, perhaps cancer or one of its precursors. In that case, perhaps 10 gm/day by IV would not go amiss. But in the case of a human, over 20 years of age and already on the exponential Gompertz-Makeham hazard curve, what then?

In that case, where precisely their internal redundancy is failing is, for most adult human individuals, inaccessible information. They know that the damage is there and that it is growing daily; that is, they know that they are aging from the time they achieve adulthood. Take the modest estimate of 22 mg per kg of body mass in a human as the total tissue store. that would be about 1500 mg in a 70 kg human. Let that be the total daily dose, taken in two to four distinct parts, with one (largest dose) preferably before the nightly rest interval. One could go even higher by taking the greater estimate of 32 mg per kg of body mass instead; that yields 2200 mg per day in a 70 kg human. The larger dose might be more appropriate for an individual human who had a more advanced age or who was clearly suffering more accumulated degenerative damage; for example, an ex-smoker. Likewise, for someone who had had poor Vitamin C status, larger doses might be indicated initially with a fall-back to a sustaining level.

This dose should be patterned around a daily cycle of exertion and rest in order to synchronize (through the concept of dynamic kinetic stability) with the flow of catabolic stress and anabolic rebuilding. That is, for moderate levels of exertion, the maximum dose (out of two or more) should be taken before rest. For more extreme levels of exertion; e.g., a marathon, then a large dose should be delivered before the event.

The primary question is whether to maximize the antioxidant effects of the vitamin during exertion or not. For normal levels of exertion, the answer would be no, in order to allow for a level of stress that would induce a training effect. For extreme exertion, (several hours at over 50% of VO2max), then protection against too much oxidative stress damage would argue in favor of high doses of antioxidants prior to exertion as well as afterwards.

In other words, consideration of an optimal dose must account for the age of the subject (to identify a likely position on the Gompertz-Makeham curve), their mass, their current levels of physical training and the daily patterns of exertion, their apparent health (or taken the other way, obvious signs of degenerative disease), and their current Vitamin C status. Levine’s recommendation of 200 mg per day of Vitamin C for provably healthy individuals at the very minimum of the Gompertz-Makeham curve has to be taken as an optimal level for individuals with that degree of mortality, or remaining redundancy. By working out to a full 13,000 mg at the most extreme age, and following the exponential hazard rates of Gompertz-Makeham, you come up with the following table:

## Optimal Daily Vitamin C Dose, by Age

Age, in yearsDose, in mg
10126
20200
30318
40506
50804
601279
702033
803233
905141
1008175
11013000

You look at your table of exponential growth in Vitamin C dosage, and considering that it matches exactly the increased need for all of the functions of the vitamin in an individual relative to their age and expected continued longevity, you are pleased. Your law is

$Dose = 79.1 exp(t/{21.56})$

For humans younger than age 10, one could go back to the life table above and increase the daily dose; but this would also have to account for the fact that a very young child’s reduced mass has not been taken into account. That is, an infant’s mass will be far less than the 70 kg that has been assumed in this analysis. Still, one could derate for mass and prorate for early childhood mortality to get a number for any age.

In contrast, if a human knows that they have certain kidney problems (renal failure), then stressing that organ with higher levels of Vitamin C or other soluble micronutrients is not appropriate.

Finally, you write your report, hit “Submit” and head home. It’s been fun, but you’re ready to get back and have some of your lovely wife’s home-made silicon chips. The processed ones they make on-board this ship just don’t taste as good. At least they keep your teeth from falling out.

# Conclusion

I have contrasted two competing ways of looking at the question of micronutrient dosage in general, and of Vitamin C in particular (as a proxy of a broad range of micronutrients). The first context is the science of inanimate objects; the second context is the science of living things.

I offered two images to crystallize these competing points of view: a bucket leaking water versus a hummingbird floating in front of a flower. The leaking bucket has no intent, no plan, no inner structure. In contrast, the hummingbird, still in the air in front of the flower, is in a state of dynamic and kinetic stability because it intends to be there.

The mission of micronutrients within the body is to protect and defend tissues from damage. In general, that damage increases at an accelerating pace as we age. So says the data and the Gompertz-Makeham model.

We ought to take micronutrients with the intent of protecting and defending our tissues from accumulated degenerative damage. We can estimate that damage, and our need for micronutrients, in terms of the Gompertz-Makeham model.

This suggests a progressive dosage of Vitamin C and similar micronutrients that elevates the daily levels of the micronutrient in proportion to the expected accumulated damage within us, with qualifications for two kinds of specific cases. If one is clearly aware of certain conditions that would ask for higher levels of Vitamin C at an earlier age; for example, if one smokes or already has a diagnosis of cancer, then higher levels may be appropriate sooner. In contrast, if one knows one has certain kidney problems (renal failure), then stressing that organ with higher levels of Vitamin C or other soluble micronutrients is not appropriate.