The mysteries of blood

November 10, 2010 § 3 Comments

Be honest — would you have guessed that red blood cells are mysterious?  No, I wouldn’t have either.  They’re the simplest cells in our bodies, for goodness sake — they don’t even have DNA.  All they do is carry hemoglobin around, picking up oxygen as they pass the lungs and gradually dumping it everywhere else.  How hard can that be to understand?  And we’ve studied them in various ways for over 450 years.

But indeed it turns out that there are significant holes in our knowledge of how the number, size and hemoglobin concentrations of red blood cells are controlled, and how these control systems go wrong in anemia.  We do know where new red blood cells come from — the bone marrow — and we know some of the factors that control the development and release of new red blood cells, such as erythropoietin.  The feedback control between “too few red cells” and “more erythropoietin needed” goes mainly through the kidneys; the mechanism the kidney uses to sense oxygen levels (protein hydroxylation) and induce erythropoietin synthesis has been an area of active research.  What we know less about is what happens to these new red blood cells once they get out in the circulation.

The population distribution of red blood cells in a healthy person is usually very stable.  But the behavior of individual cells must be quite dynamic: about a quarter of a trillion [2.5 x 10(11)] new cells are added to the population daily, and so a similar number must be taken away.   New red blood cells — which can be visualized because they carry residual RNA — are considerably larger than mature red blood cells.  They shrink by about 10% in the few days after they are released from bone marrow; then they continue to shrink, but more slowly, by shedding hemoglobin-containing vesicles.  Because red blood cells have no DNA, they can’t replace the hemoglobin lost in the vesicles, so the loss of hemoglobin is irreversible.  Calculations show, though, that they must be able to pick up membrane from somewhere; another mystery.  We don’t really know how or why the vesicle shedding happens.  And finally, after about 3-4 months, they’re cleared; and naturally, we don’t know what the trigger is for clearance (age? size? hemoglobin content?). The volume vs. hemoglobin plot for a population of red blood cells in a normal person is shown left (mature cells in red, newborn cells in blue); you can see the strong diagonal (labeled MCHC for mean corpuscular hemoglobin concentration) that is caused by the coordinated loss of volume and hemoglobin, and the surprising emptiness where the very small cells should be.

A new paper from John Higgins and L. Mahadevan (Physiological and pathological population dynamics of circulating human red blood cells, Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1012747107) asks whether, in the absence of a better mechanistic understanding of these shrinkage and death processes, modeling can provide some insights into what is going on. The brilliance of this strategy is that it taps into a huge pool of mostly unregarded data.  Hospitals do red blood cell analyses on practically everyone, sooner or later, and if they suspect you may be at risk of developing anemia they run the same analysis on you multiple times.  The standard test is called a Complete Blood Count, or CBC; it involves a wide range of measurements on both the white and red cells in your blood, including red blood cell size and hemoglobin content.  Although the distributions of red blood cell volume and hemoglobin content are routinely measured, what is generally reported to the doctor handling the patient is just an average.  The authors were especially interested in finding out whether the information on population distribution, properly analyzed, can help improve the diagnosis of anemia.

Higgins and Mahadevan took what’s known about the normal population distributions of red blood cells and developed a model of the underlying dynamics of individual cells.  They’re careful to explain that the precise form of the model they come up with is less important than the qualitative features.  Their model assumes that the average cell has two phases of shrinkage: first it shrinks rapidly towards an ideal volume/hemoglobin content ratio (the amount of shrinkage is larger the further away the cell is from the “correct” ratio), and then as it gets close to that ratio it starts shrinking more slowly, with volume and hemoglobin lost (on average) in proportion.  Both the fast and slow shrinkage involve a certain level of stochasticity; this explains the breadth of the distribution around the diagonal line that describes the ideal volume/hemoglobin content ratio. To describe the trajectory of a single red blood cell, they use a Langevin equation, and they use the Fokker-Planck equation to model the dynamics of the population (making the assumption that the system is close to steady state).  Then they introduce red cell clearance as a probability function, with the probability of clearance depending on how close the cell is to a threshold volume/hemoglobin content value.

Putting all of this together, they have a model with six main parameters.  Two parameters describe how quickly a newborn red blood cell progresses towards the ideal volume/hemoglobin content ratio; one represents the rate of change after the ideal ratio has been achieved; two describe the “scatter” or diffusion around the ratio; and one represents the threshold for cell death.  Now they can use this model to extract their 6 parameters from measurements of red blood cell populations in healthy individuals and in patients with anemia.  They looked at three different forms of anemia, each resulting from a different cause: iron deficiency anemia (primarily caused by poor nutrition), thalassemia trait (genetics), and anemia of chronic disease (inflammation).  In each case they chose individuals with mild disease, so that the steady-state assumption would not be too severely violated.

After all this build-up, you perhaps won’t be surprised when I tell you that patients with anemia do have different best-fit parameters from normal patients.  That’s interesting all by itself, but here’s what’s more interesting: the parameters that change are different in the different forms of anemia.  Thalassemia trait and iron deficiency anemia are more similar to each other than to anemia of chronic disease or normals: they shrink slower in the fast phase of shrinkage, faster in the slow phase, and, strikingly, have a much lower threshold for cell death. We already knew that the red blood cells in patients with these diseases are especially small — they’re called “microcytic” anemias for this reason — but it’s now possible to tease out the different contributions to this phenomenon.  The change in threshold for clearance is the most striking, to me; does this show that the (unknown) red blood cell clearance mechanism can be regulated by lack of oxygen?  And if so, why doesn’t this (unknown) regulatory mechanism also reduce the threshold for cell death in anemia of chronic disease patients?

Iron deficiency anemia patients also show ten-fold more variability in hemoglobin loss in the slow phase of red cell shrinkage than normals.  This may provide a way to distinguish iron deficiency anemia from thalassemia trait, which would be an important clinical contribution since mistakenly treating a person with thalassemia as if they were iron deficient can cause serious problems. Most important from the clinical point of view, this model-based analysis may allow earlier diagnosis of some forms of anemia.  Using the example of an individual diagnosed with iron-deficient anemia who had three CBC tests at two-month intervals before the diagnosis was made, Higgins and Mahadevan show that they can identify abnormal parameter changes a full two months before the anemia became clinically detectable.  The cool thing about this is that it doesn’t require a new machine, or a new test; all the data are already collected routinely, it’s just a question of running it through a new computer program.  [On the other hand, the cynical side of me suggests that a new machine that goes ping might be more marketable than mathematics.  Only time will tell.]

Higgins, J., & Mahadevan, L. (2010). Physiological and pathological population dynamics of circulating human red blood cells Proceedings of the National Academy of Sciences DOI: 10.1073/pnas.1012747107

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§ 3 Responses to The mysteries of blood

  • Dhondy says:

    Earned an editorial in the NEJM. Good work Higgins & Mahadevan!

    Would really appreciate some explanation on the Langevin & Focker Planck equations in layman’s terms if possible. Medical background, hence almost inevitably struggle with advanced Physics, but would love to understand how this works.

    • John Higgins says:

      Thanks for your interest.

      You can think of a Langevin equation as one way to describe how the volume and intracellular hemoglobin mass change over time for a single red blood cell (RBC). In general, the Langevin description is particularly useful when there’s an average or expected behavior that can be measured with high accuracy and when there are also other events that occur that we can’t predict accurately, but whose statistics we can estimate well enough. This situation is very common in medicine.

      For example, in the case of RBC maturation we can predict the average change in RBC volume very accurately by measuring the average reticulocyte volume (known as the MCVr) and comparing it to the average mature RBC volume (MCV). There are obviously other events affecting RBC volume because there is significant variation in the volume of mature RBCs. We can lump these other events together and estimate their combined effect because we know that they must lead to just enough variation in the volume reduction process to convert the initial variation in reticulocyte volume (whose coefficient of variation is often called RDWr) into the volume variation we measure in the mature RBC population (RDW).

      Using this approach we can construct a Langevin equation, but in order to decide whether this equation is accurate, we would have to follow lots of individual RBCs over the course of their lifespan and measure their volumes and hemoglobin along the way. We can’t (currently) do that, but we can make lots of snapshot measurements of populations of hundreds or more young RBCs and mature RBCs. We then need a way to describe changes for volume and hemoglobin for this entire population, not just the single RBC that the Langevin equation describes. The Fokker-Planck equation is this description. If each cell in the population behaves according to the same Langevin equation, then we can derive a Fokker-Planck equation that will describe the integrated behavior of the entire population. The Fokker-Planck equation will predict volume and hemoglobin changes for a population of reticulocytes, and we can compare its prediction to what we measure in the hospital lab.

      I hope this description helps!


  • Dhondy says:

    Many thanks, That’s extremely helpful.

    What you and Mahadevan have done will hopefully spark off a revolution in the use of mathematical modelling in Medicine. It’s something I have always dreamt of, but alas, simply lack the training in Maths/Physics to pull it off. This kind of paper sparks the dying embers of that long held fascination back to life.

    Am going through your paper in detail and hope to be able to pester you with more queries in the future as I take it all in (painfully, as is my lot, having not read Maths for many years)!

    Keep up the good work.

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