The purpose of the following analysis is to search for the viral epitopes that elicited – in a ME/CFS patient – IgGs against a set of 6 peptides, determined thanks to an array of 150.000 random peptides of 16 amino acids each. These peptides were used as query sequences in a BLAST search against viral proteins. No human virus was found. Three phages of bacterial human pathogens were identified, belonging to the classes Actinobacteria and γ-Proteobacteria. One of these bacteria, Serratia marcescens, was identified in a similar study on 21 ME/CFS cases.
Scientists have been speculating about an infectious aetiology of ME/CFS for decades, without never being able to link the disease to a specific pathogen. The idea that the disease might be triggered and/or maintained by an infection is due to the observation that for most of the patients the onset occurs after an infectious illness (Chu, L. et al. 2019). It has also been observed that after a major infection (whether parasitic, viral or bacterial) about 11% of the population develops ME/CFS (Mørch K et al. 2013), (Hickie I. et al. 2006).
In recent years, the advent of new technologies for pathogen hunting has given renewed impulse to the search for ongoing infection in this patient population. A 2018 study, investigating the genetic profile of peripheral blood for prokaryotic and eukaryotic organisms reported that most of the ME/CFS patients have DNA belonging to the eukaryotic genera Perkinsus and Spumella and to the prokaryotic class β-proteobacteria (alone or in combination) and that these organisms are statistically more present in patients than in controls (Ellis J.E. et al. 2018). Nevertheless, a previous metagenomic analysis of plasma by another group revealed no difference in the content of genetic material from bacteria and viruses between ME/CFS patients and healthy controls (Miller R.R. et al. 2016). Moreover, metagenomic analysis pursued in various samples from ME/CFS patients by both Stanford University and Columbia University has come empty (data not published, R, R).
2. Immunological methods
Another way of investigating the presence of current and/or past infections that might be specific of this patient population is to extract the information contained in the adaptive immune response. This can be made in several ways, each of them having their own limits. One way would be to collect the repertoire of T cell receptors (TCRs) of each patient and see if they have been elicited by some particular microorganism. This is a very complex and time-consuming method that has been developed in recent years and that I have described in details going through all the recent meaningful publications (R). The main limitation of this method is that, surprisingly, TCRs are not specific for a single epitope (Mason DA 1998), (Birnbaum ME et al. 2014), so their analysis is unlikely to reveal what agent selected them. On the other hand, the advantage of this method is that T cell epitopes are linear ones, so they are extremely suited for BLAST searches against protein databases. An attempt at applying this method to ME/CFS is currently underway: it initially gave encouraging results (R), then rejected by further analysis.
Another possible avenue for having access to the information registered by adaptive immunity is to investigate the repertoire of antibodies. The use of a collection of thousands of short random peptides coated to a plate has been recently proposed as an efficient way to study the response of B cells to cancer (Stafford P. et al. 2014), infections (Navalkar K.A. et al. 2014), and immunization (Legutki JB et al. 2010). This same method has been applied to ME/CFS patients and it has shown the potential of identifying an immunosignature that can differentiate patients from controls (Singh S. et al. 2016), (Günther O.P. et al. 2019). But what about the antigens eliciting that antibody profile? Given a set of peptides one’s antibodies react to, a possible solution for interpreting the data is to use these peptides as query sequences in a BLAST search against proteins from all the microorganisms known to infect humans. This has been done for ME/CFS, and the analysis led to several matches among proteins from bacteria, viruses, endogenous retroviruses and even human proteins (in fact, both this method and the one previously described can detect autoimmunity as well) (Singh S. et al. 2016). There are several problems with this approach, though. First of all, the number of random peptides usually used in these arrays is not representative of the variety of possible epitopes of the same length present in nature. If we consider the paper by Günther O.P. and colleagues, for instance, they used an array of about 10^5 random peptides with a length of 12 amino acids each, with the number of all the possible peptides of the same length being 20^12 ∼ 4·10^15. This means that many potential epitopes one has antibodies to are not represented in the array. Another important limitation is that B cell epitopes are mainly conformational ones, which means that they are assembled by the folding of the proteins they belong to (Morris, 2007), the consequence of this being that the subset of random peptides one’s serum react to are in fact linear epitopes that mimic conformational ones (they are often called mimotopes) (Legutki JB et al. 2010). This means that a BLAST search of these peptides against a library of proteins from pathogens can lead to completely misleading results.
Recently an array of overlapping peptides that cover the proteins for many know viruses has been successfully used for the study of acute flaccid myelitis (AFM). This technology, called VirScan, has succeeded in linking AFM to enteroviruses where metagenomic of the cerebrospinal fluid has failed (Shubert R.D. et al. 2019). This kind of approach is probably better than the one employing arrays of random peptides, for pathogen hunting. The reason being that a set of only 150.000 random peptides is unlikely to collect a significant amount of B cell epitopes from viruses, bacteria etc. Random peptides are more suited for the establishment of immunosignatures.
3. My own analysis
I have recently got access to the results of a study I was enrolled in two years ago. My serum was diluted and applied to an array of 150.000 peptides with a length of 16 random amino acids (plus four amino acids used to link the peptides to the plate). Residues Threonine (T), Isoleucine (I) and Cysteine (C) were not included in the synthesis of peptides. Anti-human-IgG Ab was employed as a secondary antibody. The set of peptides my IgGs reacted to has been filtered with several criteria, one of them being subtracting the immune response common to healthy controls, to have an immune signature that differentiates me from healthy controls. The end result of this process is the set of the following six peptides.
Table 1. My immunosignature, as detected by an array of 150.000 random peptides 20-amino-acid long, four of which are used for fixing them to the plate and are not included here.
The purpose of the following analysis is to search for the viral epitopes that elicited this immune response. To overcome the limitations enumerated at the end of the previous paragraph I have decided to search within the database of viral proteins for exact matches of the length of 7 amino acids. Why this choice? A survey of a set of validated B cell epitopes found that the average B cell epitope has a linear stretch of 5 amino acids (Kringelum, et al., 2013); according to another similar work, the average linear epitope within a conformational one has a length of 4-7 amino acids (Andersen, et al., 2006). To filter the matches and to reduce the number of matches due to chance, I opted for the upper limit of this length. I excluded longer matches to limit the number of mimotopes for conformational epitopes. Moreover, I decided to look only for perfect matches (excluding the possibility of gaps and substitutions) so to simplify the analysis. It is worth mentioning that a study of cross-reactive peptides performed for previous work (Maccallini P. 2016), (Maccallini P. et al. 2018) led me to the conclusion that cross-reactive 7-amino-acid long peptides might often have 100% identity.
So, to recap, I use the following method: BLAST search (blastp algorithm) against viral proteins (taxid 10239), a perfect match (100% identity) of at least 7-amino-acid peptides (≥43% query cover), max target sequences: 1000, substitution matrix: BLOSUM62.
Table 2 is a collection of the matches I found with the method described above. You can look at figure 1 to see how to read the table.
Table 2. Collection of the matches for the BLAST search of my unique set of peptides against viral proteins (taxid 10239). HP: human pathogen. See figure 1 for how to read the table.
There are no human viruses detected by this search. There are some bacteriophages and three of them have as hosts bacteria that are known to be human pathogens. Bacteriophages (also known as phages) are viruses that use the metabolic machinery of prokaryotic organisms to replicate (figure 2). It is well known that bacteriophages can elicit specific antibodies in humans: circulating IgGs to naturally occurring bacteriophages have been detected (Dąbrowska K. et al. 2014) as well as specific antibodies to phages injected for medical or experimental reasons (Shearer WT et al. 2001), as reviewed here: (Jonas D. Van Belleghem et al. 2019). According to these observations, one might expect that when a person is infected by a bacterium, this subject will develop antibodies not only to the bacterium itself but also to its phages.
If that is the case, we can use our data in table 2 to infer a possible exposure of our patient to the following bacterial pathogens: Stenotrophomonas maltophilia (HP), Serratia marcescens (HP), Mycobacterium smegmatis mc²155 (HP). In brackets, there are links to research about the pathogenicity for humans of each species. M. smegmatis belongs to the class Actinobacteria, while S. maltophila and S. marcescens are included in the class γ-Proteobacteria.
Interesting enough, Serratia marcescens was identified as one of the possible bacterial triggers for the immunosignature of a group of 21 ME/CFS patients, in a study that employed an array of 125.000 random peptides (Singh S. et al. 2016). This bacterium accounts for rare nosocomial infections of the respiratory tract, the urinary tract, surgical wounds and soft tissues. Meningitis caused by Serratia marcescens has been reported in the pediatric population (Ashish Khanna et al. 2013).
The next step will be to perform a similar BLAST search against bacterial proteins to see, among other things, if I can find matches with the six bacteria identified by the present analysis. A further step will be to pursue an analogous study for eukaryotic microorganisms and for human proteins (in search for autoantibodies).
During last summer, I’ve pursued a lot of things. I delivered a speech in Turin, after the screening of the documentary Unrest, about the OMF-funded research on the use of the measure of blood impedance as a possible biomarker for ME/CFS (video, blog post, fig. 1, fig. 2).
Then I flew to London to attend the Invest in ME conference, the annual scientific meeting that gathers researchers from all over the world who shared their latest work about ME/CFS. There I met Linda Tannenbaum, the CEO of the Open Medicine Foundation, whom I had the pleasure to encounter for the first time about a year before in Italy, and I introduced myself to Ronald Davis (fig. 3), the world-famous geneticists turned ME-researcher because of his son’s illness. I presented to him some possible conclusions that can be driven from the experimental results of his study on the electrical impedance of the blood of ME/CFS patients, with the use of an electrical model for the blood sample (R, paragraph 6).
In London, I was able to visit the National Gallery and while I was passing by all these artistic treasures without being able to really absorb them, to get an enduring impression that I could bring with me forever, I decided to sit down and to copy one of these masterpieces (I can’t draw for most of the time, and when I improve for a few weeks in summer, I usually have to carefully choose where to put my energies). I sat probably beside one of the least important portraits collected in the museum (Portrait of a young man, Andrea del Sarto, figure below) and I started copying it with a pen. When I finished, the museum was closing, so that I missed all the works by Van Gogh, among many other things.
We were at the beginning of June, I was experiencing my summer improvement, a sort of substantial mitigation of my illness that happens every other summer, on average. But because of these travels, I elicited a two-month worsening of symptoms, during which I had to stop again any mental and physical activity: I just lay down and waited. At the beginning of August, I started thinking and functioning again and I almost immediately decided to quit what was my current project (a 600-page handbook of statistics that I commenced in 2017) and I started studying mathematical modelling of enzymatic reactions (figures 4 and 5).
I knew that these reactions were described by ordinary differential equations and that I could solve them numerically with the methods that I studied just before I got sick, about 18 years ago. I was interested in the metabolic trap theory by Robert Phair, an OMF-funded researcher. So I downloaded a chapter of one of the most known books of biochemistry and a thesis by a Turkish mathematician on metabolic pathways simulation and I started my journey, working on the floor (I have orthostatic intolerance even when I get better in Summer, so I can’t use a desk, figure 6). I ended up learning the rudiments of this kind of analysis, also thanks to a book by Herbert Sauro and to some suggestions by dr. Phair himself! Some of the notes I wrote in August are collected here.
At the beginning of September, I was absorbed by the problem of how to study the behaviour of the steady states of tryptophan metabolism in serotoninergic neurons of midbrain as the parameters of the system change. This kind of analysis is called bifurcation theory and I literally fell in love with it… In figure 6 you can also see a drawing: I was drawing a picture I have been thinking about for the last 20 years. It is a long story, suffice it to say that in 1999, just before my mind faded away for 18 months, I started studying the anatomy of a man who carries a heavy weight on his back (see below). That was my first attempt of communicating what was happening to me, of describing my disease.
Only recently I considered to not represent the weight, which is a more appropriate solution since this is a mysterious disease with no known cause, and I made a draft (the one in figure 2) that I then used as a starting point for the drawing below. I finished this new drawing at the beginning of September, in a motel room of San José, in California, just in time for donating it to Ronald Davis (figures below) when I moved to the US to attend the third Community Symposium at Stanford (see here). In California, many surprising things happened: I met again Linda Tannenbaum and Ronald Davis, and yes, I encountered also Robert Phair! But this is another story…
In the following pictures, you can see how the drawing evolved. Notably, the figure in the centre changed his face and some part of his anatomy. The three figures are meant to be a representation of the same figure from three different points of view. It is more like a project for a sculpture, a monument that is much deserved by these patients.
At Stanford, I had the chance to be face to face with one of my preferred sculptures ever: The Thinker, by Rodin, in both its version: the model moulded first, on the top of The Gates of Hell, and the big one (crafted later), now considered the iconic symbol of Philosophy, but likely originally meant to be a metaphor for creative thinking (I say that because the original sculpture included in The Gates of Hell is a representation of the Italian poet Dante Alighieri, depicted in the act of imagining his poem).
At the end of September, my mind started fading away again. I knew that would have happened, even though I had an irrational hope that this year would have been different. At that point, I was in Italy and I asked some friends to help me organize a trip to the southern hemisphere, in order to live another summer. It required more time than I would have hoped. I am going to leave from Italy only tomorrow. My goal: Argentina. I have been able to do something, at a highly reduced speed, in October, though. I have developed a model for solar radiation at sea level, in function of the day of the year, of the latitude, and of the distance from the Sun (I have considered the actual elliptic orbit of our planet). The main problem has been the modelling of absorption and of diffusion of radiant energy from our star by the atmosphere, but I solved it. Part of these notes are here, but I want to self-publish the end product, so I keep the rest to myself. In that period, I was also able to find the exact solution of the improper integral known as the Stefan-Boltzmann law, something I tried to do in the summer of 2008, in vain, in one of my recovery-like periods. In figure 6 you can see one of the results of my model for solar radiation: the monochromatic emissive power at sea level in function of the day of the year, for the city of Buenos Aires.
My intention was to use that model to choose the perfect place where to move in order to have environmental conditions that closely resemble the ones that we have in Rome from June to September (the period in which my improvements happen). I also wanted to quantitatively study the effect of both infrared radiation and ultraviolet radiation on my biology. There are several interesting observations that can be made, but we will discuss these subjects another time, also because I had to quit this analysis given my cognitive deterioration. The video below is a byproduct of the geometric analysis that I had to pursue in order to build my model for solar radiation at sea level.
Dawn and dusk at a latitude of 42 degrees north, during three years of the silent rolling of the Earth on its silken ellipse. Three years of adventures, suffering, joy and death.
So, by November my mind was completely gone and my physical condition (namely orthostatic intolerance and fatigue) had worsened a lot. This year I have been able to try amphetamines: I had to go from Rome to Switzerland to buy them (they are restricted drugs that can’t be sold in Italy and can’t be shipped to Italy either). One night I felt good enough to take a train to Milan and then to take another transport to the drug store. And back. I managed to do the travel but I pushed my body too far and I had to spend the following month in bed, 22 hours a day, with an even worse mental deterioration. It is like having a brain injury. Amphetamines have been useless in my case, despite two studies on their potential beneficial effect in ME/CFS.
Right now, I am collecting all the books and the papers that I need with me in Argentina (figure above), in case I will improve enough to study again. But what am I going to work on?
I want to finish my model of solar radiation, with some notes on the effect of infrared radiation, ultraviolet radiation and length of the day on the immune system. There is a mathematical model published recently that links the length of the day to the power of the innate immune system, and I want to write a code that calculates the relative activity of the innate immunity in function of latitude and day of the year. I would like to self- publish it as a booklet.
I want to finish my handbook of statistics.
I need to correct a paper submitted for publication (it has been accepted, but some corrections have been required).
I want to deepen my understanding of the bifurcation theory for metabolic pathways and to continue studying tryptophan metabolism with this new knowledge.
I want to complete my work on autoantibodies in ME/CFS (see this blog post) and to submit it to a journal. I have been working on that for a while, inventing new methods for the quantitive study of autoimmunity by molecular mimicry.
Should I improve again in Argentina, several avenues can be followed in order to understand the reason why summer causes this amelioration in my own case. I have many ideas and I’ll hopefully write about that in the future. Of course, I also want to read all the new research papers I have missed in the last months. I will bring with me my handbook of anatomy for artists because I hope to be able to draw again, and I won’t miss this opportunity to leave some other handcrafted images behind me for posterity, that can’t care less, obviously! I would really like to finish the drawing below because I feel that in this draft I have found a truly elegant (and mechanically correct) solution for the hip joint of a female robot.
Now I am useless, my mind doesn’t work and I am housebound. I can’t read, I can’t draw, I can’t do calculations, I can’t do coding, I can’t cook… This has been the quality of my life for most of the last 20 years. This is a huge waste: I would have used these years to perform beautiful and useful calculations and to pursue art. I would really make people understand how tragic this disease is in its cognitive symptoms, what we lose because of it. This is, in fact, the reason behind this blog post: I wanted to give an idea of what I can do when I feel better, and of what I would have done if there had been a cure.
I have lost most of my adult life, but I will never accept to waste a day without fighting back.
All the following studies have been made mainly thanks to private funding. Please, consider a donation to the Open Medicine Foundation, in order to speed up the research. See this page for how to donate to OMF.
OMF Scientific Advisory Board Director Ronald W. Davis, PhD, has just delivered a speech about ME/CFS at the Albert Einstein College of Medicine at Columbia University in New York. In what follows you find several screenshots that I have collected during the lecture, accompanied by a very short description. I imagine that a video will be soon made available but in the meantime let’s take a look at these slides.
Indoleproprionate is reduced in ME/CFS patients. This molecule is not produced by our own metabolism, it comes from a bacterium of the gut (Clostridium sporogenes) which is low in patients. It has a neuroprotectant effect. Indoleproprionate is currently used in some clinical trials for other diseases and it might be available in the next future as a drug.
Hydroxyproline is high and this is believed to indicate collagen degradation. Ron Davis talked about the case of a ME/CFS patient who turned out to have a problem in the craniocervical junction which was fixed by surgery. Is there a link between high hydroxyproline and abnormalities of the joints (the neck among them) that some patients seem to have?
The increase in electrical impedance in blood samples (as measured by the so-called Nanoneedle device) only happens when cells from ME/CFS patients are incubated with plasma from patients. When these same cells (white blood cells) are incubated with plasma from healthy donors, the impedance is normal. For an introduction to this experiment, click here. The published work is here.
The Nanoneedle study has been extended with 20 more patients and 20 more controls. This device can be used for drug screening, other than for diagnosis.
The peptide called Copaxone, now used in Multiple Sclerosis, seems to work in reducing the impedance in the nanoneedle device (click on the images to enlarge). Suramin also has some effect (on the right). It doesn’t seem as good as Copaxone though, to me…
SS-31 is an experimental drug for the mitochondrial membrane. It does work when used in the nanoneedle device! (click on the images to enlarge).
Nailbed capillaroscopy could be a new instrument for ME/CFS diagnosis. Inexpensive and already in use in hospitals.
No new or known pathogen has been found in patients, so far. This project is still in progress. It is updated as new technologies for pathogen hunting become available.
All the severe patients have at least one defective copy of the IDO2 gene. The same applies to 46 additional ME/CFS patients that have been recently tested for this gene. This is a common genetic problem in the general population, but it is ubiquitous in these patients. And a statistically significant difference is thus present between ME/CFS patients and healthy controls. This discovery has lead to the development of the metabolic trap hypothesis, which has been recently published (here). For an introduction, read this blog post of mine. They are planning to test the metabolic trap hypothesis in vivo using cellular cultures!
Patients have high mercury (maybe from fish in their diet) and low selenium in hair. Low selenium can reduce the conversion of T4 to T3 in the liver. Low T3 might be a cause of fatigue. High uranium was also detected!
All these studies have been made mainly thanks to private funding. Please, consider a donation to the Open Medicine Foundation, in order to speed up the research. See this page for how to donate to OMF.
I disabili possono costituire una fonte di reddito cospicua per chi lavora nell’ambito della assistenza a queste persone. Per le disabilità riconosciute, infatti, il sistema sanitario offre (giustamente) dispositivi costosi e personale con le più varie mansioni. Ma in molti casi appalta questo genere di servizi.
Come funziona un appalto? Si redige un capitolato di gara, che elenca i requisiti che i candidati devono possedere per poter partecipare alla gara. Questo capitolato viene reso pubblico solo al momento del bando. Il vincitore sarà l’ente che, tra i candidati, meglio soddisfa i requisiti del capitolato di gara.
Un anno fa, dopo una indagine di vari mesi, la Guardia di Finanza di Bologna ha iscritto nel registro degli indagati quattro dipendenti Ausl e due rappresentanti della Associazione Italiana Assistenza Spastici (AIAS Bologna) con l’accusa di aver truccato un appalto. Infatti il file del capitolato di gara sarebbe stato corretto più volte – prima ancora del bando – dai rappresentanti di AIAS Bologna, in modo da cucirsi addosso il concorso e assicurarsi la vittoria , .
Si parla di due milioni e 130 mila euro circa, in tre anni, con possibile rinnovo per altri tre anni. Si tratta della gara per il Centro Regionale Ausili (che include anche il Centro Ausili Tecnologici) e per il Centro Adattamento Ambiente Domestico. La gara è stata sospesa , .
Dopo ulteriori accertamenti, il procedimento contro uno dei dipendenti Ausl è stato archiviato, ma per il resto l’indagine è ancora in piedi, che io sappia .
Io stesso sono il beneficiario dei servizi di una di queste aziende appaltatrici, che assistono le persone con disabilità. E non potrei fare a meno di questi servizi, non sopravviverei.
Proprio per questo mi infastidisce non poco venire a sapere di tali illeciti (presunti, fino a che il procedimento giudiziario non termina il suo corso). E perché sono spiacevoli e pericolosi questi illeciti? Perché in fondo suggeriscono che le disabilità più remunerative sono quelle a cui si presta più attenzione e a cui si offrono più servizi.
Non sarebbe necessario scrivere una nota all’articolo pubblicato dall’Osservatorio Malattie Rare (O.ma.r.) sulla malattia di Lyme alcuni giorni fa , in cui si liquidano i possibili esiti cronici della patologia in parola come non esistenti o riconducibili ad altre patologie, “non ultime le patologie psichiatriche”. Non sarebbe necessario, ho scritto, perché coloro che si occupano professionalmente della malattia di Lyme, nonché i pazienti, sono consapevoli che in realtà le possibili sequele della infezione acuta da Borrelia burgdorferi sono ben documentate in letteratura e costituiscono un problema di enorme portata su cui si concentra oggi molta ricerca, spesso di alto profilo .
E allora perché questa preterizione? Perché un articolo divulgativo pubblicato da O.ma.r. ha notevole risonanza e – se incompleto o impreciso – rischia di alimentare false credenze tra i neofiti. L’articolo in questione è encomiabile nel divulgare nozioni preziose sulla fase acuta della malattia e nel mettere in guardia contro test e trattamenti non provati, ma è del tutto fuorviante nella parte dedicata alle sequele croniche (penultimo paragrafo).
La malattia di Lyme risponde alle attuali cure antibiotiche nel 90% dei casi. Questo significa che un decimo di coloro che ogni anno, in estate, contrae la malattia per il morso di un vettore (principalmente l’Ixodes ricinus in Europa) andrà incontro a una condizione cronica (cioè con durata superiore ai sei mesi) e debilitante, nota in seno alla comunità scientifica con il nome di post-treatment Lyme disease syndrome, PTLDS , . Il nome scelto (che si potrebbe tradurre come Sindrome della malattia di Lyme dopo trattamento) sta a indicare che i pazienti sperimenteranno sintomi, nonostante i trattamenti della fase acuta.
La causa della PTLDS è al momento non nota. Alcuni studi supportano l’ipotesi che ci sia una disfunzione immunitaria in questi pazienti. Due studi hanno dimostrato la presenza di anticorpi contro il sistema nervoso centrale nella metà dei pazienti PTLDS , , solo per citare i più recenti. Per approfondimenti su questo argomento si legga qui e qui. La ricerca in questo campo continua .
Altri gruppi hanno dimostrato la persistenza del patogeno – dopo trattamento – sia nel modello animale della malattia di Lyme , , , che negli esseri umani , . Presso la Columbia University è stata avviata una raccolta di tessuti provenienti da individui deceduti, che abbiano avuto una ben documentata infezione da Borrelia burgdorferi (link) proprio per indagare ulteriormente questo aspetto.
Numerosi sforzi e investimenti sono stati profusi recentemente nella ricerca di nuovi agenti antimicrobici per questa infezione, nella speranza di scongiurare le sequele croniche, da parte della Università di Stanford , , della Università Johns Hopkins , , , della Università Northeastern .
Nell’articolo di O.ma.r si legge che la sindrome in questione – chiamata impropriamente post-Lyme dall’autore – è caratterizzata da “sintomi soggettivi – e dunque non quantificabili – quali affaticabilità e difficoltà a concentrarsi”. A questo proposito è quasi superfluo ricordare che molti sintomi sono soggettivi, finché l’ingegneria non ci offre uno specifico strumento di misura: si pensi alla risonanza magnetica nella sclerosi multipla o all’elettroencefalogramma nella epilessia. In secondo luogo, questi sintomi sono solo in parte soggettivi, infatti i pazienti PTLDS presentano alterazioni quantificabili nel sistema immunitario , , nella espressione genica , nel metabolismo , etc.
Per quanto riguarda il riferimento all’ambito psichiatrico, vale la pena fare delle considerazioni. E’ senz’altro vero che le principali patologie psichiatriche (i disturbi dell’umore da un lato e le psicosi dall’altro) contemplano gli episodi infettivi come fattore di rischio , , tuttavia la PTLDS semplicemente non si sovrappone alle patologie psichiatriche, né nella presentazione clinica, né nella epidemiologia, per questo è una categoria nosografica a sé stante (vedi qui). Per fare un esempio concreto: negare l’evidenza, come sembra fare questo articolo, è una delle possibili manifestazioni della schizofrenia paranoide , ma né questo tipo di sintomi né altri sintomi patognomonici per malattie psichiatriche sono menzionati nella definizione di caso della PTLDS .
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Recently there have been some anecdotal reports of patients with a diagnosis of ME/CFS who met the criteria for a diagnosis of craniocervical instability (CCI). After surgical fusion of this joint, they reported improvement in some of their symptoms previously attributed to ME/CFS (R, R). After some reluctance, given the apparently unreasonable idea that there could be a link between a mechanical issue and ME/CFS, I decided to look at this avenue. So here I am, with this new blog post. In paragraph 2, I introduce some basic notions about the anatomy of the neck. In paragraph 3, I describe three points that can be taken from the middle slice of the sagittal sections of the standard MR study of the brain. These points can be used to find four lines (paragraph 4) and these four lines are the basis for quantitative diagnosis of craniocervical instability (paragraph 5-10). In paragraph 11, I describe CCI. In paragraph 12, I discuss the possible link between craniocervical instability and ME/CFS. In paragraph 13, there is a collection of measures from the supine MRIs of some ME/CFS patients. In the last paragraph, I propose an alternative definition of CCI, with the introduction of Euler’s angles.
2. Basic anatomy
The craniocervical (or craniovertebral) junction (CCJ) is a complex joint that includes the base of the skull (occipital bone, or occiput), the first cervical vertebra (atlas or C1), the second cervical vertebra (axis or C2), and all the ligaments that connect these bones (Smoker WRK 1994). This joint encloses the lower part of the brainstem (medulla oblongata) and the upper trait of the spinal cord, along with the lower cranial nerves (particularly the tenth cranial nerve, the vagus nerve). Since the CCJ is included in the series of sagittal sections of every MR study of the brain, its morphology can be easily assessed (figure 1, left). It is worth mentioning that the CCJ is the only joint of the body that encloses part of the brain. The atlas and the axis are represented with more detail in figure 1 (right), where their reciprocal interaction has been highlighted. From a mechanical point of view, these two bones make up a revolute joint, with the rotation axis going through the odontoid process. This is only a simplification, though, because while it is true that the atlantoaxial joint provides mainly axial rotation, there are also 20 degrees of flexion/extension and 5 degrees of lateral bending, which means that spherical joint would be a more appropriate definition. Other degrees of freedom are provided at the level of the occipital atlantal joint, where 25 degrees of motion are provided for flexion/extension, 5 degrees of motion are provided for one side lateral bending and other 10 degrees are provided for axial rotation (White A. & Panjabi M.M. 1978).
The measurement of the Grabb’s line and of the clival-canal angle is based on a simple algorithm which starts with the identification of three points on the midline sagittal image of a standard MRI scan of the head (figure 2). In order to find this particular slice, search for the sagittal section where the upper limit of the odontoid process reaches its highest and/or the slice with the widest section of the odontoid process. This algorithm is mainly taken from (Martin J.E. et al. 2017). In looking at T1-weighted images, always keep in mind that cortical bone (and cerebrospinal fluid too) gives a low signal (black strips) while marrow bone gives a high signal (bright regions) (R).
Clival point (CP). It is the most dorsal extension of the cortical bone of the clivus at the level of the sphenooccipital suture. This suture can’t be seen clearly in some cases (figure 3 is one of these cases). So another definition can be used for CP: it is the point of the dorsal cortical bone of the clivus at 2 centimetres above the Basion (see next point).
Basion (B). It is the most dorsal extension of the cortical bone of the clivus. This is the easiest one to find!
Ventral cervicomedullary dura (vCMD). This is the most dorsal point of the ventral margin of the dura at the level of the cervicomedullary junction. I find this point the most difficult to search for and somehow poorly defined, but this is likely due to my scant anatomical knowledge.
Posteroinferior cortex of C-2 (PIC2). It is the most dorsal point of the inferior edge of C2.
Connecting the three points found in the previous paragraph allows us to define four lines (figure 3) that will be then used to calculate the Grabb’s measure and the clival-canal angle.
Clival slope (CS). It connects CP to vCMD. It is also called the Wackenheim Clivus Baseline (Smoker W.R.K. 1994).
Posterior axial line (PAL). It connects vCMD to PIC2.
Basion-C2 line (BC2L). It connects B to PIC2.
Grabb’s line (GL). It is the line from vCMD that is orthogonal with BC2L.
We now know all we need in order to take two of the most important measures for the assessment of craniocervical junction abnormalities.
5. The clival-canal angle and its meaning
The clival-canal angle (CXA) is the angle between CS and PAL. The value of this angle for the individual whose scan is represented in figure 4 is 142°. This angle normally varies from a minimum of 150° in flexion to a maximum of 180° in extension (Smoker WRK 1994). Ence, what we should normally see in a sagittal section from an MR scan of the brain is an angle between these two values. A value below 150° is often associated with neurological deficits according to (VanGilder J.C. 1987) and it is assumed that a CXA below 135° leads to injury of the brainstem (Henderson F.C. et al. 2019). A clival canal angle below 125° is considered to be predictive of CCI according to (Joaquim A.F. et al. 2018). In a study on 33 normal subjects employing standard MRI, CXA was measured in the sagittal section of each subject: this group had a mean value of 148° with a standard deviation of 9.88°; the minimum value was 129° and the maximum one was 175° (Botelho R.V. et al. 2013). The reader may have noted that the mean CXA in this study is below the cutoff for neurological deficits according to the 1987 book. This might be due to the fact that there is a difference between the measure taken on an MRI sagittal section and the one taken on radiographic images.
It has been demonstrated with a mathematical model that a decrease in the clival-canal angle produces an increase in the Von Mises stress within the brainstem and it correlates with the severity of symptoms (Henderson FC. et al. 2010). Von Mises stress gives an overall measure of how the state of tension applied to the material (the brainstem in this case) causes a change in shape. For those who are interested in the mathematical derivation of this quantity (otherwise, just skip the equations), let’s assume that the stress tensor in a point P of the brainstem is given by
Then it is possible to prove that the elastic potential energy due to change in shape stored by the material in that point is given by
where E and ν are parameters that depend on the material. Since in monoaxial stress with a module σ the formula above gives
by comparison, we obtain a stress (called Von Mises stress) that gives an idea of how the state of tensions contributes to the change of shape of the material:
In the brainstem, this parameter – as said – appears to be inversely proportional to the clival-canal angle and directly proportional to the neurological complaints of patients, according to (Henderson FC. et al. 2010). For a complete mathematical discussion of Von Mises stress, you can see chapter 13 of my own handbook of mechanics of materials (Maccalini P. 2010), which is in Italian though.
6. The Grabb’s measure and its meaning
The Grabb’s measure is the length of the segment on the Grabb’s line whose extremes are vCMD and the point in which the Grabb’s line encounters the Basion-C2 line. In figure 4 this measure is 0.8 centimetres. This measure has been introduced for the first time about twenty years ago with the aim of objectively measuring the compression of the ventral brainstem in patients with Chiari I malformation. A value greater or equal to 9 mm indicates ventral brainstem compression (Grabb P.A. et al. 1999). In a set of 5 children with Chiari I malformation and/or basal invagination (which is the prolapse of the vertebral column into the skull base) a high Grabb’s measure was associated with a low clival canal angle (Henderson FC. et al. 2010). When using MRI, it is assumed that values above 9 mm is abnormal (R) but I have not been able to find statistical data on this measure in MRIs of healthy individuals. Moreover, the study by Grabb was mainly on a pediatric population (38 children and two adults) with Chiari malformation. So it is unclear if these measures can be used to assess the CCJ in adults. The measure was made on sagittal sections of MRIs.
The CXA only takes into account osseous structures (it depends on the reciprocal positions between the body of the axis and the clivus), so it can potentially underestimate soft tissue compression by the retro-odontoid tissue. This problem can be addressed with the introduction of the Grabb’s measure (Joaquim A.F. et al. 2018). Nevertheless, we can assume that they both measure the degree of ventral brainstem compression, and if you look at figure 3 you realize that as the angle opens up, the Grabb’s measure becomes shorter. Points and lines described in these paragraphs for two more patients are represented in figure 4.
7. Horizontal Harris measure
Another measure that has been introduced to check the anatomical relationship between the skull and the Atlas is the distance between PAL and point B (figure 5). This measure has been introduced in (Harris J.H. e al. 1993) where it was performed in 400 adults and with a normal cervical spine and in 50 healthy children. In the first group, 96% of the individuals had a distance of the basion from PAL longer than 1-4 mm and shorter than 12 mm. All the children had a distance below 12 mm. This measure has been used recently to assess craniocervical instability in hypermobile patients (Henderson F.C. et al. 2019), along with the Grabb’s measure and the clival-canal angle. We will refer to this measure as HHM. It is important to mention that the study by Harris was based on radiographs, so it is unclear if they can be used for a comparison of measures taken from MRI sagittal sections. Yet a measure below 12 mm was considered normal in a study employing MRI (Henderson F.C. et al. 2019).
8. Distance between Chamberlain’s line and the odontoid process
Another measure that has been introduced to determine whether occipitovertebral relationship is normal or not is the distance between the Chamberlain’s line and the closest point of the tip of the odontoid process (also called dens) (figure 6). The Chamberlain’s line extends between the posterior pole of the hard palate and the posterior margin of the foramen magnum (called opisthion) (Smoker W.R.K. 1994). In a study on 200 healthy European adults employing standard MRI, this measure was taken from the T1 weighted sagittal section of each subject. Measures start from the cortical bone, i.e. from the dark signal. The mean was -1.2 mm with a standard deviation SD = 3 mm (Cronin C.G. et al. 2007). The minus before the number indicates that the mean position of the selected point of the dens is below the line.
9. Distance between McRae’s line and the odontoid process
McRae’s line is drawn from the anterior margin of the foramen magnum (basion) to its posterior border (opisthion). It was introduced in 1953 to assess normality at the level of the CCJ (McRae D.L. et Barnum A.S. 1953). The distance between McRae’s line and the closest point of the tip of the dens can be used, as in the case of Chamberlain’s line, to assess abnormality of the CCJ along the z-axis (figure 7). In a study on 200 healthy European adults employing standard MRI, this measure was taken from the T1 weighted sagittal section of each subject. Measures start from the cortical bone, i.e. from the dark signal. The mean was -4.6 mm with a standard deviation SD = 2.6 mm (Cronin C.G. et al. 2007). The minus before the number indicates that the mean position of the selected point of the dens is below the line. In normal individuals, the dens is always below the McRae’s line (McRae D.L. et Barnum A.S. 1953), (Cronin C.G. et al. 2007).
10. Distance between basion and odontoid process
It is the distance between the basion and the tip of the dens. It is also called basion-dental interval (BDI) and it has been proposed that a value greater of 10 mm is abnormal and predicts occipito-atlantal instability. Moreover, the average value is 5 mm, according to (Handerson F. 2016). I have not been able to find statistical data for BDI measured in MRI sagittal sections of healthy subjects. Moreover, I do not have a cutoff for the minimum value.
11. Craniocervical instability
According to some authors, the craniocervical junction is considered to be unstable (craniocervical instability, CCI) in the case of “any anomaly that leads to neurological deficits, progressive deformity, or structural pain”. A clival canal angle below 125° and/or a Grabb’s measure above 9 mm are considered to be predictive of CCI (Joaquim A.F. et al. 2018). Craniocervical instability has been described in congenital conditions like Down syndrome (Brockmeyer D 1999), Ehlers-Danlos syndrome (Henderson F.C. et al. 2019), and Chiari malformation (Henderson FC. et al. 2010) as well as in rheumatoid arthritis (Henderson F.C. et al. 1993).
In one study on craniocervical junction stabilization by surgery in five patients with Chiari I malformation or basal invagination (Henderson FC. et al. 2010), inclusion criteria, beside abnormal Grabb’s measure and CXA, were:
signs of cervical myelopathy (sensorimotor findings, hyper-riflexia);
signs of pathology at the level of the brainstem, collected in this table;
severe head and/or neck pain, improved by the use of a neck brace for at least a 2 weeks period.
The same inclusion criteria were adopted in another similar study on patients with hereditary hypermobile connective tissue disorders (Henderson F.C. et al. 2019).
Several mechanisms are believed to play a role in the genesis of the clinical picture described in CCI: stretch of the lower cranial nerves (vagus nerve is among them) and of the vertebral arteries; deformation of the brainstem and of the upper spinal cord (Handerson F. 2016).
12. Craniocervical instability and ME/CFS
CCI has been described in Ehlers-Danlos syndrome hypermobile type (Henderson F.C. et al. 2019), although the prevalence of CCI in EDSh has not been established, yet (to my knowledge). At the same time, an overlapping between EDSh and ME/CFS has been reported in some studies: most of EDSh patients met the Fukuda Criteria, according to (Castori M. et al. 2011) and it has been proposed that among patients with ME/CFS and orthostatic intolerance, a subset also has EDS (Rowe P.C. et al. 1999), (Hakim A. et al. 2017). So, it might seem not unreasonable to find CCI in a subgroup of ME/CFS patients.
Moreover, both in CCI and in ME/CFS there is an involvement of the brainstem. Briefly, hypoperfusion (Costa D.c: et al. 1995), hypometabolism (Tirelli U. et al. 1998), reduced volume (Barnden L.R. et al. 2011), microglial activation (Nakatomi Y et al. 2014), and loss of connectivity (Barnden L.R. et al. 2018) have been reported in the brainstem of ME/CFS patients. Basal ganglia dysfunction has also been documented in ME/CFS (Miller AH et al. 2014), and this could be an indirect measure of midbrain abnormal functioning, given the connection between substantia nigra (midbrain) and basal ganglia, via the nigrostriatal tract. It is worth mentioning here that vagus nerve infection has been proposed as a feasible cause of ME/CFS (VanElzakker MB 2013) and vagus nerve (the tenth cranial nerve) has its origin in the lower part of the brainstem. Recently, brainstem pathology in ME/CFS (midbrain serotoninergic neurons alteration, in particular) has been theorized as part of a mathematical model on disrupted tryptophan metabolism (Kashi A.A. et al. 2019), (R). So, one might argue that CCI could in some cases lead to a clinical picture similar to the one described in ME/CFS because in both these conditions there is a pathology in the same anatomical district (figure 8).
We know that in most of the cases ME/CFS starts after an infection (Chu L. et al. 2019). That said, how could CCI be linked to this kind of onset? The presence of CCI in rheumatoid arthritis (Henderson F.C. et al. 1993) might be a clue for a causal role of the immune system in this kind of hypermobility. In fact, a link between hypermobility and the immune system has been found also in a condition that is due to the duplication/triplication of the gene that encodes for tryptase (a proteolytic enzyme of mast cells) (Lyons JJ et al. 2016).
A piece of evidence against a link between CCI and ME/CFS is perhaps represented by the results of a study on EDSh patients with CCI who underwent surgery for their craniocervical junction abnormalities. Before surgery, all the 20 patients reported fatigue among their symptoms and two yers after surgery the improvement in this symptom was not statistically significant, despite improvement in the craniocervical joint measures (CXA and Grabb’s measure) and improvement in overall functioning (Henderson F.C. et al. 2019). This seems to be a clue against the role of CCI in fatigue, at least in this patient population.
13. Craniocervical measures in a few ME/CFS patients
I have collected standard MRIs of the head of seven ME/CFS patients and I have performed the measures described in this article, using the sagittal section of T1 weighted series. Data are collected in table 1.
GM stands for Grabb’s measure and the cutoff for this value has been taken from an MRI study on children with Chiari malformation (Grabb P.A. et al. 1999). I have not been able to find a study on adult normal subjects, so I don’t have any reliable statistical data on that measure. Yet, the reported cutoff of 9 mm is what is commonly indicated for GM (R), (Handerson F. 2016), (Joaquim A.F. et al. 2018). HHM stands for horizontal Harris measure and the cutoff was deduced from (Henderson F.C. et al. 2019), but again, I have not found statistical data on this measure from MRIs sagittal sections of an adult healthy population. BDI is the basion-dens interval and the cutoff comes from (Handerson F. 2016) and no statistical data available on a suitable population. CDD and MDD are the distances of the tip of the dens from the Chamberlain’s line and the McRae’s line, respectively and I got the statistical data from an MRI study on adult healthy subjects (Cronin C.G. et al. 2007). CXA is the clival-canal angle: statistical data were from an MRI study on 33 healthy adults (Botelho R.V. et al. 2013), while the cutoff was indicated in (Henderson F.C. et al. 2019).
The only abnormal values found are the distance between the tip of the dens and both Chamberlain’s line and McRae’s line in P2 and the Grabb’s measure in P7, with the caveat that I don’t have suitable statistical data for comparison, in the latter case. And of course, I don’t know what the meaning of these slightly abnormal values is. Of notice, none of these patients would fit the criteria proposed in (Henderson F.C. et al. 2019) for surgery of the craniocervical junction.
Patient 4 should probably be excluded from this table: she had a documented B12 deficiency at the onset of her disease; she was treated with vitamin B12 injections. After some months she has substantially improved. So it might have been a case of vitamin B12 deficiency. She also has a problem with iron, which tends to be low and has to be supplemented; since vitamin B12 and iron are both absorbed in the small intestine, this patient may have some pathology in that area. In fact, signs of inflammation were found in a sample of her duodenum, but it was not possible to define a specific diagnosis (celiac disease was ruled out, as well as Crohn’s disease). Interesting enough, this patient had a diagnosis of POTS (by tilt table test) and vitamin B12 deficiency has been linked to POTS (Öner T. et al. 2014). As mentioned, she is in remission now.
Let’s try now a statistical analysis for the values of the clival canal angle reported in Table 1, using as control group the one published in (Botelho R.V. et al. 2013). We can use Cantelli’s inequality (see Eq. 2, paragraph 15) and extend it to a random vector. We get for the p value:
In our case m = 8, µ = 148, σ = 9.88. By using the following very simple code we calculate a p value < 0.03, which is statistically significant. The problem here is that the measure of the CXA in the control group has been made by someone else than me, so this might be a source of error. Moreover, the sample is very small. All that said, a tendency towards a reduction of the clival canal angle among ME/CFS patients might be further proof of increased mobility of the cranio-cervical joint in this patient population, in agreement with previous studies on other joints (Rowe P.C. et al. 1999), (Hakim A. et al. 2017).
mu = 148
ds = 9.88
m = 8
p = 1.;
x = [142, 146, 142, 142, 135, 140, 140, 139];
p = p*( 1/( 1 + ( ( (mu-x(i))/ds )^2 ) ) );
14. Craniocervical instability and Euler’s angles
A more sound definition of CCI might perhaps be obtained with the introduction of the angles that are used to describe the orientation of a rigid body with respect to a fixed coordinate system. To simplify our analysis, we assume here that atlas (C1) and axis (C2) are fixed one to the other. Then, consider the coordinate system (O; x, y, z) in figure 1 to be fixed to C1-C2 and then let’s introduce a second coordinate system (Ω; ξ, η, ζ), fixed to the skull. The orientation of (Ω; ξ, η, ζ) with respect to (O; x, y, z) is given by the angles ψ, φ, θ, called Euler’s angle (figure 7). The angle θ is the one between z and ζ. In order to define the other two angles, we have to introduce the N axis, known as line of nodes, which is the intersection between plane xy and plane ξη. That said, ψ is the angle between x and N, while φ is the angle between ξ and N.
All that said, craniocervical hypermobility may be defined as follows.
Def. We have CCI when there is an increase in the physiological range of Euler’s angles and/or when |ΩO|≠0.
In this definition, we have assumed that in physiological conditions the length of the vector ΩO is nought. The length of ΩO is indicated as |ΩO|. The condition |ΩO|≠0 means that at least one of the components of ΩO along the axises x, y, z is different from zero.
The reader can easily recognize now that:
the clival-canal angle is a measure of instability in the angle θ; we can also say that clival-canal angle measures instability around N;
Grabb’s measure and Horizontal Harris measure both indicate instability along the x-axis; they are a measure of the x component of vector ΩO;
Chamberlain’s line gives a measure of instability along the z-axis; the same applies to McRae’s line and to BDI.
15. Cantelli’s inequality
To assess the statistical significance of the experimental data in Table 1 we have used Cantelli’s inequality, also known as one-tailed Chebyshev’s inequality. Given the random variable X whose distribution has mean E[X] and variance Var[X], then Cantelli’s inequality states that:
for any η>0. The importance of these two inequalities is that they are true whatever the distribution is. In the case of our patient’s MRS data, we only knew mean values and standard deviations (which is the square root of variance) of the distributions of the metabolic values of the control group. So one way to assess significance was to use this inequality (the other way would be to use the less precise Chebyshev’s inequality). To prove Eq. 1 and Eq. 2 we have first to prove Markov’s inequality, which states that
for any a>0. In order to prove that, consider that for the probability on the left of the inequality we can write
At the same time, the expectation (or mean) of the distribution can be written
Thus we have
and Markov’s inequality is proved. Let’s now come back to the proof of Cantelli’s inequality. If we consider the random variable Y = X – E[X] we have that P(Y≥η) = P(Y+t≥η+t) and assuming that η+t > 0 we have
That said, Markov’s inequality gives
For the expectation on the right we have
and knowing that E[Y²] = Var[X] and that E[Y] = 0, we can write
The function on the right of the inequality is represented in Figure 4. It is easy to recognize that it assumes its lower value for t = Var(X)/η and this proves Eq. 1. The other inequality (Eq. 2) can be proved in the same way, considering the random variable Z = E[X] – X.
In this article, I report on the results from two research groups in which different experimental settings were used to measure electric impedance in blood samples from ME/CFS patients vs healthy controls. One of these studies comes from Stanford University and has been just published in PNAS: it is freely available here. The other one has been presented by Alan Moreau during the NIH conference on ME/CFS, and it is unpublished (R). In paragraph 2 I introduce the definition of impedance, in paragraph 3 you will learn something about the electric behaviour of cells, in paragraph 4 there is a description of the device used by the Stanford University group, in paragraph 5 there are the results of the experiment from Stanford University, in paragraph 6 there is a discussion of these results, in paragraph 7 the results from the other group are reported, and these two studies are compared in paragraph 8. In paragraph 9 I reported on two drugs that have shown the promise to be of therapeutic use in ME/CFS. Other notes follow in the last two paragraphs. If you are not interested in technical details on impedance (or if you don’t need them), go directly to paragraph 5.
In this paragraph, I try to give a very simple and short introduction to circuits in a sinusoidal regime in general, and to impedance in particular. The main definition that we need, for that purpose, is the so-called Steinmetz transform for a sinusoidal function. Let’s consider the sinusoid
where A is called amplitude and is the maximal value that the function can reach, ω is the angular frequency (also called pulsatance) which is an indication of how fast the value of the function changes in time, α is the phase and it gives the indication of what the value of the function a(t) was for t = 0. The Steinmetz transform consists of the univocal association of the sinusoid a(t) with the complex number
also called phasor (which stands for phase vector), where j=√(-1) is the imaginary unity. A complex number can then be easily represented as a vector in the complex plane (see figure 1).
Let’s now consider the elementary circuit in figure 2 (which is also a simplified model of the device in the study by Ron Davis), where a generator of electrical potential is linked to another circuit (depicted as a box in the figure, on the left) that in our case is represented by the sample of peripheral blood white cells incubated in plasma. But it could be an arbitrarily complex net made up of conductors and what follows would still hold. Let’s assume that the electric current and the voltage of the generator are given respectively by
We can associate to these sinusoids their respective phasors with the Steinmetz transform, which gives
That said, we define impedance of the sample, the complex number that we obtain dividing the phasor of u(t) by the phasor of i(t):
Impedance describes several physical properties of the box in figure 2. Without going into details (this is beyond the scope of this article) just consider what follows.
The real part of impedance represents the resistance of whatever is inside the box of figure 2, which can be seen as its ability to transform electric energy into heat, i.e. kinetic energy at a molecular level. The higher the value of the resistance, the more the ability to generate heat.
The imaginary part of the impedance (called reactance) can be positive or negative. When it is positive it indicates the ability of whatever is inside the box to translate a magnetic field into voltage. The higher the positive reactance, the more its ability to generate a voltage from a magnetic field. A positive reactance is also called inductive reactance.
When reactance is negative, it means that whatever is inside the box, it has the ability to store energy in an electric field: the higher the absolute value of the reactance, the more the energy stored in an electric field within the box. A negative reactance is also called capacitive reactance.
No matter how complex the system in the box is, its external electrical behaviour is completely characterized by its impedance, which means that the system can also be simplified in a series of an electrical component whose only relevant property is a resistance equal to the real component of impedance, and a second component completely characterized by a reactance with a value equal to the imaginary component of impedance (figure 2, on the right).
3. Impedanceof cells
The study of the impedance of cellular cultures is a field that started probably in the early nineties. In a paper from the Rensselaer Polytechnic Insititute (NY), it was demonstrated that the measure of electrical impedance of a single cell layer was more sensitive than optical microscopy for the measure of changes of nanometers in the cell diameter or subnanometer changes in the distance between the cell layer and the electrodes (Giaever I. & Keese CR. 1991). In that pivotal paper, a mathematical model for the impedance of a layer of cells was also proposed and solved, but it is beyond the scope of this article. A simplified electrical model of a cell layer is provided by a parallel of a capacitance due to dielectric properties of the cell membrane, and a resistance due to the cell membrane, to the cytoplasm and to the layer between cells (Voiculescu I. et al. 2018). We can add a resistance for the solution in which cells are incubated and we obtain the circuit in figure 3.
Remember now that the only electrical property that we can directly measure is the total impedance (both the real component and the imaginary one). So we have to find the relationships between these two components and the physical parameters introduced in figure 3. For the equivalent impedance of the sample (see the last paragraph for the mathematical passages) we have:
The dependence of the real part of Z_cl and of its imaginary component to R_cl and C_cl can be got from figure 4. The absolute value of Z_cl is represented in figure 5.
The capacitance in this formula is due – as said – to the dielectric properties of the plasma membrane. We can see a cell as a spherical capacitor, where two conductive layers (one in the cytoplasm and the other one in the extracellular space) are separated by the outer membrane. The insulating portion of a phospholipid membrane is of about 4.5 nm and it has been found that the capacitance per square cm of the cell membrane is one μF (Matthews GG, 2002). Since the permittivity constant ε is known, we can calculate the dielectric constant κ of a lipid membrane quite easily (see the last paragraph), and we find κ=5.
4. The nanoneedle
The device used for the measurement of the impedance of blood samples from ME/CFS patients is an array of thousands of sensors. Each sensor is made up of two conductive layers, separated by a dielectric material (figure 6). Each sensor is a sinusoid circuit that operates at a frequency of 15 kHz and at a voltage with an amplitude of about 350 mV. In figure 6, I have added the electric scheme for the circuit made up by the sensor itself and the sample, according to what seen in the previous paragraph. I have added some resistances and capacitors for the electrodes, according to (Esfandyarpour R et al. 2014).
As you can see from the picture, one of the dimensions of the sensor is below one micron, while the other is of about 3 microns. Keep in mind that the diameter of the average white blood cell is of about 15 microns… To me, such a small size makes it difficult the application to this system of both the electrical model by Ivar Giaever and Charles Keese and of the simplified one presented in the previous paragraph, which have been designed to describe the behaviour of a layer of cells that grow above an electrode that can harbour many cells on its surface. And in fact, in their paper, Esfandyarpour R. and his colleagues have sketched a different model (R, B), even though they haven’t used it to draw any conclusion or interpretation from the experimental data, yet.
5. The experiment
The measurement of the impedance of samples from ME/CFS patients and controls has been made with an array of thousands of electrodes, each one like the one in figure 6. The system took 5 measures of impedance for second and the experiment on each sample lasted for about 3 hours. The researchers measured, for each point in time, both the real and the imaginary component of the impedance of the sample. They also measured the module of the impedance.
Each sample consisted of peripheral blood mononuclear cells (PBMC) incubated in patient’s own plasma (plasma is blood without erythrocytes, platelets and white blood cells), at a concentration of 200 cells per μL. It might be useful to remember that PBMCs are basically all the white blood cells that are present in peripheral blood but granulocytes, which have multi-lobed nuclei and, as such, are not “monuclear”.
The researchers drew blood from 5 severe patients, 15 moderate patients (diagnosed by a physician according to the Canadian Consensus Criteria) and 20 healthy controls, with 5 of them age- and gender-matched to 5 of the ME/CFS patients.
About 20 minutes were required for the impedance to reach a steady state (the baseline level, characterized by swings in impedance below 2% of its value). The measures for each sample have been divided by the value of impedance at the baseline. This is the reason why the baseline has a value of 1 in the diagrams. After the steady state was reached, the researchers added 6 μL of NaCl to the samples. After a transient reduction in impedance, the samples from healthy controls returned to the baseline value. In samples from patients, the initial reduction in impedance after NaCl introduction was followed by a dramatic change in both the real component and the imaginary component of impedance. The normalized real part, in particular, had an increase of 301.67% ± 3.55 (see figure 7 and R).
6. What does it mean?
In the experiment by Stanford University, they added NaCl to the samples and this likely led to the activation of the sodium-potassium pump that requires a molecule of ATP in order to transport 3 Na ions outside the cells (and two K ions inside). This would be the only way for these cells to maintain the correct intracellular concentration of sodium, pumping out those Na ions that found their way to the cytoplasm from the plasma. This is like putting a cell on a stationary bike. What the experiment says is that this effort made by the cells to maintain homeostasis leads to huge changes in the electrical properties of the samples from ME/CFS patients, while producing virtually no changes in the samples from healthy controls. But what is the origin of the change in impedance?
If we consider the electrical model that I have proposed in figures 3 and 6 and looking at figure 4 (left), we might hypothesise that the change comes from a reduction in the capacitance C_cl which is due to the dielectric properties of cell membranes. A change in composition in these membranes could lead to a reduction in C_cl and thus to the observed increase in the real component of the total impedance. This might perhaps be linked to the reduction in the metabolism of the main components of the plasma membrane (sphingolipid, phospholipid and glycosphingolipid) in patients vs controls previously reported in a metabolomic study (Naviaux R et al. 2016). A reduction in the dielectric properties of cell membranes could also explain the increase in the module of impedance observed in this study (see figure 5). But it is worth noting again that the model I used for the description of the electrical properties of the sample is a hugely simplified version of the one proposed in (Giaever I. & Keese CR. 1991) and it has been developed for electrodes that are many times larger than the one used by Esfandyarpour R and colleagues. As said elsewhere, the authors have proposed a different, more complex, electric circuit (R, B) and they wrote that the process of using it to interpret the experimental data is currently on-going. But they did note that a change in plasma membrane composition might be responsible for the observed change in impedance, in one point of the article, among other possible explanations.
A release of molecules (cytokines?) from the PBMCs into the plasma might also be the cause of the change in impedance, but if we assume that our model in figure 3 is reliable, these molecules would only change the value of R_su, so the imaginary component of the impedance would not be affected, while we know that there is a change in that component too. But again, our model is a very simplistic one.
A change in the shape or size of the cells would lead to a change in C_cl. But the authors observed the samples in standard live microscopy imaging and they were not able to record any significant cell size difference in samples from ME/CFS patients vs samples from healthy controls.
7. Canadian impedance
During the NIH conference on ME/CFS, the Canadian group led by Alan Moreau, presented, at the end of a speech about microRNAs, a measure of impedance on immortalized T cells incubated with plasma from healthy controls, plasma from ME/CFS patients, and plasma from patients with idiopathic scoliosis (figure 8) and, as you can see, there is an increase in impedance with the increase in plasma concentration only in the second group (R). This measure has been made with the CellKey system, after stimulation of cells with G-coupled protein receptors agonists (Garbison KE et al. 2012). It is also worth mentioning that this impedance is the one due to the flow of charges in the extracellular space and that it seems to be the module of impedance, rather than the real or the imaginary part.
Alan Moreau also noted that if we subgroup ME/CFS patients according to differences in circulating microRNA, we find that plasma from two of these groups leads to an increase in impedance while plasma from three other groups induces a decrease in impedance, if compared with T cells incubated with plasma from healthy controls (figure 9).
8. The X factor
Even though the Canadian experiment is not directly comparable to the one from the Stanford University group, nevertheless it is a partial confirmation of that result. Moreover, since in the Canadian experiment the cells are the same for all the groups (it is a line of immortalized T cells) and what changes is only the plasma they are incubated in, we can say that the origin of the electrical shift in these samples is something that is present in the plasma of patients (an X factor) and it might be due to the interaction between this X factor and cells. This interpretation is in agreement with a previous observation from a Norwegian group who incubated muscular cells in serum from 12 patients and from 12 healthy donors: they found an increase in oxygen consumption and in lactic acid production in cells incubated with sera from patients vs cells incubated with sera from healthy controls. This experiment was performed using the Seahorse instrument (Fluge et al. 2016). It is worth noting that in this case only serum was used, and serum is plasma without clotting factor.
The idea of an X factor present in plasma (or serum) of patients is even more suggestive if we take into account the unpublished result presented by Ron Davis during the NIH conference, called the “plasma swap experiment”, performed with the nanoneedle device (R). As you can see from figure 10, the increase in impedance happens only when cells are incubated with plasma from ME/CFS, no matter whether the cells are from healthy controls or from ME/CFS patients.
It is extremely important here to note that several filtrations of the plasma from patients have been made by the Stanford Group in order to discover what the X factor is: they have concluded that it is neither a metabolite nor a cytokine. Alan Moreau noted also that it is probably not an antibody. It turned out that it might be an exosome, a vesicle released by cells which contains – among other molecules – microRNA molecules. As Ron Davis said: “I guess that the signal is coming from damaged mitochondria, but it is only a guess” (R).
9. Drug testing
The authors of the study on the nanoneedle device are interested in using it for drug testing. Ron Davis reported during the last Emerge Australia conference (R) that two compounds are able to reduce the alteration in impedance seen in PBMCs incubated with plasma from patients: Copaxone, a peptide currently used in the treatment of multiple sclerosis, and SS31, a molecule not available yet, that can scavenge mitochondrial reactive oxygen species (ROS), thereby promoting mitochondrial function (Escribano-Lopez I. et al. 2018), (Thomas DA et al 2007).
10. Limitations of the study from Stanford University
Even though the differences observed in the electric properties of the samples from ME/CFS patients vs controls, after the addition of the osmotic stressor, are striking, there are some potential limitations that ought to be mentioned.
Only 5 of the 20 healthy controls were age and gender-matched to 5 ME/CFS patients. So the difference observed might be due, at least in part, to age or gender.
The difference in impedance might be due to some consequence of the disease, like deconditioning, since the healthy control was not a sedentary one.
I presented the content of this blog post after the screening of Unrest in Turin (Italy) in May 2019 (video in Italian).
11. Mathematical notes
The calculation of the impedance Z_cl of the sample (figure 3) is as follows:
Then you have to add the resistance R_su to the real part and you obtain Z_tot. In order to calculate the dielectric constant of the lipid membrane just follow these passages:
In order to choose the range of variation for C_cl and R_cl in the diagrams in figures 4 and 5, I calculated the capacitance of a cell, assuming a spheric shape, a radius of 5 μm, a capacitance for square cm of 1 μF, a thickness of the plasma membrane of 4.5 nm, and a dielectric constant κ=5. This gives
I then found the value of the imaginary component of the impedance of a culture of yeast cells measured by the nanoneedle, which is 800 kΩ and I set the angular frequency at 2π·15 kHz (which is the frequency of the generator of voltage of the nanoneedle). Then we have a reference value for resistance too:
The simple code (Matlab) that I used to plot the diagrams in figure 4 and 5 is the following one.
% file name = impedance
% date of creation = 4/05/2019
% we define the angular frequency
w = 2*pi*15*(10^3)
% we register the array of the capacitance axis (pico Farad)
c_span = 4.;
delta_c = c_span/30.;
n_c = c_span/delta_c;
% we register the array
c(1) = 0.;
for i = 2:30+1
c(i) = c(i-1) + delta_c;
% we define the array of resistance (mega Ohm)
r_span = 9.;
delta_r = r_span/30.;
n_r = r_span/delta_r;
r(1) = 0.;
for i = 2:30+1
r(i) = r(i-1) + delta_r;
% we register the array of the real part and of the imaginary part of impedance and its module
Rcl = r(j)*(10^6);
Ccl = c(i)*(10^(-12));
Z_r (i,j) = Rcl/( 1 + ( (Rcl^2)*(w^2)*(Ccl^2) ) );
Z_i (i,j) = (-1)*( w*Ccl )/( ( 1/(Rcl^2) ) + (w*Ccl)^2 );
Z_m (i,j) = sqrt( (Z_r (i,j)^2)+(Z_i (i,j)^2) );
% we plot the real part of the impedance
mesh(r(1:n_r), c(1:n_c), Z_r(1:n_c,1:n_r));
ylabel('capacitance (pico Farad)');
xlabel('resistance (Mega Ohm)');
legend('Real part of Impedance',"location","NORTHEAST");
% we plot the imaginary part of the impedance
mesh(r(1:n_r), c(1:n_c), Z_i(1:n_c,1:n_r));
ylabel('capacitance (pico Farad)');
xlabel('resistance (Mega Ohm)');
legend('Imaginary part of Impedance',"location","NORTHEAST");
mesh(r(1:n_r), c(1:n_c), Z_m(1:n_c,1:n_r));
ylabel('capacitance (pico Farad)');
xlabel('resistance (Mega Ohm)');
legend('Module of Impedance',"location","NORTHEAST");
Quello che segue è un frammento proveniente dagli appunti per un romanzo che tentavo di scrivere nel 2006. Il romanzo non fu mai compiuto, anche lui vittima della malattia. Ma alcuni anni dopo utilizzai parte del materiale già scritto per una storia breve, intitolata “Il flauto di Turk” (disponibile qui). Altri passaggi di quegli appunti, come questo, reclamano da anni una vita propria.
Sono circa le tre del mattino e mi trovo nel corridoio del reparto di Seconda Medicina, seduto sotto la statua della Vergine, l’unico caso di donna mediorientale con le sembianze di una biondissima scandinava. Nel pomeriggio di ieri un paio di persone sono venute davanti a questa statua per pregare, per chiedere con tutta probabilità la guarigione di un congiunto. Fissavano la statua, questa dea bellissima vestita di azzurro, divinità e fata. Dal piedistallo lei guardava in basso, congelata nell’accenno di un abbraccio irraggiungibile. Guardava in basso, un punto indefinito del pavimento, lungo una direzione difficile da intercettare, con un sorriso di porcellana, il sorriso di una di quelle vecchie bambole dal viso dipinto.
Mi è sembrato che queste due persone se ne siano andate insoddisfatte, forse infastidite da un simulacro così palesemente falso, un personaggio di plastica dei giochi delle bambine. Ma forse mi sbaglio, ho proiettato sui loro visi la mia indifferenza nei confronti di una religiosità che in fondo usa esattamente lo stesso repertorio iconografico dei culti di ogni tempo e di ogni luogo: Zeus, Apollo ed Era hanno cambiato solo il nome e il sentimento religioso ha mantenuto sempre gli stessi connotati poiché l’Umanità in fondo è rimasta la stessa: il gesticolare dei sacerdoti è ancora il rituale dello sciamano mentre le parole rivoluzionarie di quell’uomo che mendicava il pane, in Galilea, sono finite intrappolate in formule da ripetere a memoria.
Queste riflessioni mi riportano a ciò che è successo ieri, al motivo per cui sono finito qui. Allora metto un foglio su un tavolino solitario, abbandonato a pochi metri dalla dea, e inizio a scrivere, a ricostruire quello che è successo, in compagnia di una statua fredda, ben diversa da quella che ieri mi ha travolto.
Uscito dalla Facoltà di Ingegneria mi sono diretto verso la Chiesa di San Pietro in Vincoli, che sorge adiacente all’edificio universitario, in fondo a un piccolo largo. Ho attraversato la piazzetta sguarnita, da cui si innalza una breve rampa di scale, sormontata da un portico buio. Circondato da grate scure e fitte, che lo preservano da un rapporto diretto con la luce del giorno, esso costituisce tutto ciò che si scorge del luogo di culto dall’esterno. Oltre il colonnato del portico mi sono lasciato alle spalle l’atmosfera tiepida e leggermente ventilata di questi giorni, per ritrovarmi avvolto da un’aria stantia e umida, da una luce malata, indebolita e smembrata dal diaframma della trama metallica e dei pilastri. Quell’ambiente rappresenta una camera di decompressione, una tappa necessaria per sfumare il passaggio dal contingente, che ci lasciamo alle spalle, all’eterno, che ci aspetta terribile oltre le piccole porte lignee, consumate da un fiume secolare di mani.
Porte piccole per accedere a un locale molto ampio: l’improvviso cambio di scala è un espediente efficace, usato nei luoghi di culto per sopraffare il fedele, per nutrire il suo smarrimento, il suo sentimento di inadeguatezza e di miseria. In fondo il nucleo geometrico delle nostre chiese è ispirato a una religiosità antica, pagana, che si inoltra profondamente nella memoria delle generazioni sepolte. Nel ruotare di un’anta si manifesta dinnanzi a me tutta l’ampiezza della navata centrale, in parte nascosta dal buio e dunque, per quello che si potrebbe definire “effetto Leopardi”, ancora più smisurata, quasi infinita. Davanti ai miei piedi trovo una vasta superficie di pavimento assolutamente libero – i primi scranni sono molti metri dinanzi – senza punti di riferimento; sopra il mio capo si svela un cielo improvvisamente altissimo.
Conosco il potere suggestivo di questa sapiente gestione dello spazio e della luce, eppure ciò non basta a evitare che mi senta toccato, con addosso un disagio vago, una vertigine. Mi affretto verso una navata laterale, per godere del sollievo dato da un ambiente più a misura d’uomo. Le navate laterali servono a questo in fondo: sono un percorso protetto, una corsia di emergenza, un posto tranquillo dove sottrarsi alla sfacciata ostentazione di grandezza dell’Onnipotente.
Ero nella navata destra dunque, ma non ero tranquillo, perché sapevo che qui, in questa chiesa, la navata destra non è un posto dove potersi sottrarre alla voce dell’Eterno. Ho camminato col cuore in gola, lungo il corridoio scandito dalle ombre delle colonne, che tagliano il chiarore pallido dei lucernai sperduti nelle altezze della navata di mezzo. Luci e ombre. Vogliamo dire che lo spazio architettonico in cui mi trovavo aspiri a essere una metafora della nostra vita mortale? È così. L’esistenza stagna a volte in torbide paludi di oscurità dove molti purtroppo indugiano per sempre; per altri c’è invece la rinascita che nella sua commovente bellezza ripaga di ogni pena. Diciamo allora che il percorso che facevo, sotto gli sguardi pietosi dei santi, voleva essere un augurio, una metafora piena di speranza: luce, ombra, luce…
Pellegrino umile, incerto, mi sono avvicinato alla fiamma potente imprigionata nel contesto angusto e modesto del monumento funebre a papa Giulio Secondo. Con lo sguardo basso, cercando il coraggio di guardare il terribile vortice di marmo, avvolto in una poderosa spirale, ho posato le mani e gli occhi sulla balaustra, unica, misera protezione offerta al visitatore contro il tremendo gigante.
Anche senza osservarlo ne ho avvertito la presenza, nell’aria. Al suo cospetto l’atmosfera vibra in ogni molecola, come se fosse scossa dalle più basse note di un organo titanico. Un parapetto sottile, alto non più di un metro. Che protezione può dare contro la forza di quel corpo maestosamente avvitato, di quelle membra massicce?
Ho cominciato ad alzare il viso, con molta prudenza. Radici tenaci di una quercia millenaria appaiono le falangi del piede disumano, che scarica al suolo un peso grandioso. La tibia è vigorosamente avvolta da fasci nodosi e solidi di muscoli. Vedo statica forza, resa dinamica dal panneggio di una veste essenziale, dal tessuto spesso, pesante. Articolazioni robuste, tendini come corde navali, rete di turgidi vasi sanguigni: ho osservato mani come quelle stringere pale nei cantieri, o cime sui bastimenti. Riposano enormi, come il corpo massiccio, fissato nel marmo, dell’Ercole Farnese.
Il torace, immortalato in una vigorosa maturità, alimenta gli omeri spaventosi, gli avambracci, duri rami di un ampio tronco di noce. La barba è una vertigine che riempie lo spazio, agitandolo, creando flussi potenti.
Mi sono fatto animo e ho affrontato il capo del mostro, completamente dimentico ormai del motivo per cui ero lì, delle vicende di quella mattina; fuori dal tempo, oltre me stesso. La fronte corrucciata, sporgente e ossuta, è l’icona della volizione. Lo sguardo emerge infuocato dalla sinistra oscurità delle orbite profonde, incorniciato da zigomi spigolosi e larghi, siberiani.
Il visitatore è indegno dell’attenzione del gigante, è solo un’insignificante e passeggera perturbazione dell’eternità, dimensione alla quale egli appartiene. Personificazione della volontà, Mosè esige l’inchino del destino stesso.
Io dico che quel viso è riconosciuto da tutti come famigliare, da qualunque nazione si provenga, qualunque sia l’etnia e la cultura di appartenenza. È un’immagine terribile che traspare dalle acque dei ricordi degli anni più teneri, quelli in cui il piano del tavolo della cucina segnava i limiti della nostra statura, del nostro universo tattile. È il volto del papà, contrariato per quella cornice che abbiamo rotto mentre giocavamo con la palla. Del padre, il gigante delle meraviglie, dei disegni favolosi eseguiti per noi, mentre ci teneva sulle ginocchia; delle ombre fatte con le mani, sul muro della nostra cameretta, per farci addormentare.
Il Mosè incarna quest’uomo. O meglio, il suo lato oscuro, il suo potere di terrorizzare, solo con la minaccia di una punizione. Il suo potere assoluto. Mosè: il padre, il gigante padrone del mondo, poeta e demiurgo. E orco.
Ma la vita è contorta, fin nella sua più intima fibra, come le eliche che archiviano il progetto di ciascuno di noi, nelle cellule. Michelangelo questo lo sapeva e non poteva nasconderlo, neanche in questo suo monumento alla forza.
Ero perduto davanti a quella magnifica epifania, quando comincio a penetrare nel mistero che giace dietro alla prorompente carica vitalistica della statua. Ho una rivelazione sconvolgente: il gigante è impotente, ripiegato su se stesso. La sua forza si disperde tutta nell’attrito prodotto dai vortici che lo avvolgono e dalla tensione della sua torsione. Egli non riesce ad andare da nessuna parte, si agita violento, ma non conclude nulla. È il corpo di un serpente a cui è stato mozzato il capo: soffoca se stesso avvitando le spire, sfogando tutta la sua energia vitale in una terribile agonia.
Dietro la divinità avevo scoperto la miseria dell’uomo. Mi ha investito allora una marea potente di angoscia e mi sono ritrovato io stesso nell’incubo dello scultore. Il terrore di non poter concludere positivamente il mio lavoro, di non riuscire a dare un senso alla mia vita, mi stava soffocando. Oppressione toracica, angoscia indefinita, palpitazioni. La vista si stava sfocando, come dietro a un velo di lacrime. Con un’emissione impercettibile di fiato ho sussurrato:
Tutti i fiumi scorrono verso il mare e il mare non si empie mai; sempre i fiumi tornano a fluire verso il luogo dove vanno scorrendo. Ogni discorso resta a mezzo, ché l’uomo non riesce a concluderlo.
Era la Bibbia, lo sconvolgente libro di Qohèlet.
Cosa ci fa un rubicondo tedesco sopra di me? Sta sventolando una cartina sul mio viso, mentre versa nella mia bocca una bevanda fresca, con un sapore intenso. Sono disteso supino sulla scalinata davanti alla chiesa, al centro di un gruppetto di turisti teutonici che mi fissano preoccupati. Io sono confuso e debolissimo e istintivamente bevo dalla lattina poggiata sulle mie labbra, mentre guardo il viso paonazzo che mi incoraggia premuroso, accennando un sorriso. Bevo diligentemente, un sorso alla volta, mentre qualcuno mi sostiene il capo. Bevo, bevo. Ma cosa? Sento la testa dolermi e girami. Per Giove è birra! Razza di …
Non ho neanche la forza di protestare, inoltre ormai è troppo tardi. Sorrido al mio salvatore e ricado nel buio. Io non tollero l’alcool, sono totalmente astemio.
Quando rinvengo, mi ritrovo in un letto d’ospedale, con indosso un camice indecente, aperto sulla schiena. Mi hanno detto che sarò dimesso domani, dopo il giro dei medici.
In 1999 I was wandering in Pisa with a booklet in the pocket of a worn coat, too short for my frame. That coat was dark blue on the outside, green and red inside, with one mended sleeve and an austere cowl: I was much like a young monk, with his holy book (figure 1). I can remember neither its title nor the author, though. It was an introduction to statistical thermodynamics, with beautiful figures, a coloured cover, and less than 100 pages. It contained the work by Maxwell on the kinetic theory of ideal gasses, along with other material. I borrowed it from the University Library because I was fascinated by the way in which Maxwell was able to describe the properties of gasses with just a few hypotheses and some relatively easy mathematical passages. I felt that there was an enormous attraction in these methods, I realized with pleasure that math could give the power to completely understand and hold in hand physical systems and even, I started speculating, biological ones.
My second favourite composer back then was Gustav Mahler (the favourite one being Basil Poledouris): he represented my own way to classical music and I chose him because he wasn’t among the musicians my father and my brother shared a love for. I felt, during my teens, that I had to find my private space, and I met it one day on a used book stand: a cassette tape of Das Lied von The Erde, with a few sheets containing the translation to Italian of the songs. Mahler was born in 1860, a few weeks after Maxwell published his pivotal work about ideal gasses in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (R) (figure 2). But in 1999 I was repeatedly listening to a CD with a collection of songs sung by Edith Piaf and Charles Trenet, because I was studying French, and I was having a hard time with pronunciation. So, imagine a secular monk in his prime who listens to old French songs while keeping one hand on a book of statistical thermodynamics hidden in his pocket, wherever he goes, wandering in the streets of Pisa, a city which gave birth to Galileo Galilei. This seems a beautiful story, much like a dream, right? Wrong.
I had already started my struggle against the mysterious disease that would have completely erased my life in the following years. In the beginning, it had a relapsing-remitting course, so I could sometimes host the hope that I was recovering, only to find myself caught by the evil curse again. At the end of the year 1999, I was losing my mind, I knew that and I was also aware that my holy book couldn’t save me. I clearly remember one evening, I was walking on Ponte di Mezzo, a simple and elegant bridge above the Arno, and I felt that I couldn’t feel sorrow for the loss of my mind: I realized that not only the functions of my brain assigned to rational thinking were gone, but my feelings couldn’t function properly either. In fact, I noted without a true manifestation of desperation that I had lost my emotions. One day, after spending in vain about eleven hours on a single page of a textbook of invertebrate palaeontology, I accepted that I couldn’t read anymore, at least for the moment.
Had I known for sure that I wouldn’t have recovered in the following twenty years, I would have quite certainly taken my own life, jumping from a building; a fate that I have been thinking about almost every day ever since. I considered this possibility during the endless sequence of days in which there has been nothing other than the absence of my thoughts.
The distribution of velocities of an ideal gas and the one hundred years gap
In the already mentioned paper by Maxwell, he derived the probability density of the speed of the molecules of a gas, granted that the three components of the vector of speed are independent random variables (hyp. 1) and that they share the same density (hyp. 2), let’s say f. Moreover, the density of the speed has to be a function only of its module (hyp. 3). These three hypotheses together say that there is a function Φ such that
This is a functional equation (i.e. an equation in which the unknown is a function) whose solution is not detailed in Maxwell’s work. But it can be easily solved moving to polar coordinates (see figure 3) and deriving with respect to θ both members (the second one gives naught since it depends only to the distance from the origin).
Another way to solve the functional equation is to use the method of Lagrange’s multipliers, searching for the extremes of the density of the velocity, when its module is fixed. In either case, we obtain the differential equation:
which leads to the density for each component of the speed:
where σ can’t be determined using only the three hypotheses mentioned above. Considering then the well-known law of ideal gasses (pV=nRT) and an expression for p derived from the hypothesis that the collisions between the molecules of gas and the container are completely elastic, Maxwell was able to conclude that:
where m is the mass of the molecule of gas, T is the absolute temperature and K_B is the Boltzmann’s constant. It was 1860, Mahler’s mother was going to deliver in Kaliště, Charles Darwin had just released his masterpiece “On the origin of species”, forced to publish much earlier than what he had planned because of the letter he had received from Wallace, in which he described about the same theory Darwin had been working on for the previous 20 years. In the same point in time, Italy was completing its bloody process of unification, with the Mille expedition, led by Giuseppe Garibaldi.
But the functional equation I have mentioned at the beginning of this paragraph has brought with it a mystery for years, until 1976, when an employee at General Motors Corporation published a short note in the American Journal of Physics [R] in which he showed how Maxwell’s functional equation is, in fact, an example of the well known Cauchy’s functional equation:
In order to prove that, you just have to consider the following definition:
The name of the mathematician who made this observation is David H. Nash, and he has the merit of finding something new in one of the most known equation of physics, an equation that is mentioned in every book of thermodynamics, an equation that has been considered by millions of students in more than a century. It was 1976, my mother was pregnant with my brother; Alma, Gustav Mahler’s wife, had died about ten years before.
Module of random vectors
Once Maxwell found the density of probability for each component of the speed of the molecules of an ideal gas, he searched for the density of the module of the speed. There is a relatively simple way of doing that. With the following notation
we have that the repartition function of Z is given by the integrals of the density of X within the sphere in figure 4. We have:
The second expression is the same as above but in polar coordinates. Then we can obtain the density of Z by derivation of the repartition function. And this method can be extended to an m-dimensional space. This was the method used by Maxwell in his paper. And yet, there is another way to obtain the expression of the module of a random vector: I have explored it in the last months, during the rare hours in which I could function. By the way, only in the Summer of 2007 I was able to study the kinetic theory by Maxwell, eight years after I borrowed the holy book. Such a waste.
The hard way towards the density of the module of a random vector
When a student starts studying statistics, she encounters a list of densities: the normal distribution, the gamma distribution, the exponential distribution etc. Then there are several derived distributions that arise when you operate sums, roots extractions etc. on random variables. In particular, if f_X is the densty of X and Y = X², then we have
On the other hand, if Y = √X we have
Another important result that we have to consider is that given
By using these results I have been able to find that the expression of the density for the module of an m-dimensional random vector is:
In particular, for m = 3 we have
The caseof normal random vectors: the modified Bessel function
In particular, if the random vector has dimension 3 and its components are normal random variables with the same expected value and variance, we have that the density of its module is given by
where I_0 is the modified Bessel function, which is one solution of the differential equation:
whose name is modified Bessel equation. The integral expression of the modified Bessel function is:
I have coded a script in Matlab which integrates numerically this function (available here for download) which plots the surface in figure 5 and also gives the following table of values for this function.
The following is the flowchart of the script I coded.
The caseof normal random vectors with a naught expected value: the upper incomplete gamma function
If we consider random variables with an average that is zero (this is the case with the components of speed in ideal gasses), then the density is given by
which is a Chi distribution with 3 degrees of freedom, scaled with a scale parameter given by s = 1/σ. In the expression of the repartition function, ϒ is the lower incomplete gamma function, which is defined as follows:
I have written a code for its numerical integration (available here for download), the output of which is in figure 6.
Conclusion, twenty years later
The density of the module of the velocity of the molecules of an ideal gas is, in fact, a scaled Chi distribution with 3 degrees of freedom, and it is given by
It can be numerically integrated with the following script I made for Octave/Matlab, which gives the plot in figure 7. Another similar script gives the plot in figure 8. These plots represent the Maxwell-Boltzmann distribution, the centre of the holy book that an unfortunate boy was carrying in his pocket, all alone, some twenty years ago. He could have easily died by his own hand in one of the several thousand days of mental and physical disability that he had to face alone. Instead, he has survived. Had it been only for finding the Maxwell-Boltzmann distribution following another path, it would have been worth it. But he has found much more, including a bright girl, the magnificent next stage of evolution of human beings.
% file name = legge_Maxwell-Boltzmann_2
% date of creation = 22/02/2019
% it plots the density and the distribution function for the
% Maxwell-Boltzmann distribution considered as a function of temperature
% and speed
% we define some parameters
K_B = 1.381*10^(-23) % Boltzmann's constant
m = 4.65*10^(-26) % mass of the molecule N_2
% we define the array of temperature from 0° C to 1250° C
T (1) = 273.15
for i = 2:1:1250
T (i) = T (i-1) + 1.;
% it defines f_gamma in 3/2
f_gamma = sqrt(pi)/2.
% delta of integration
delta = 1.0
% it defines the array for the abscissa
z (1) = 0.;
for i = 2:1:2500
z (i) = z(i-1)+delta;
% it defines the density
for j = 1:1:1250
% it defines a constant
c = ( m/(K_B*T(j)) );
for i = 1:1:2500
f (j,i) = ( c^1.5 )*sqrt(2./pi)*( z(i)^2. )*( e^( -0.5*c*z(i)^2. ) );
% it calculates the ripartition function for He
F (j,1) = 0.;
F (j,3) = F (j,1) + delta*( f(j,1) + ( 4*f(j,2) ) + f(j,3) )/3;
F (j,2) = F (j,3)*0.5;
F (j,k+2) = F (j,k)+delta*( f(j,k)+( 4*f(j,k+1) )+f(j,k+2) )/3;
% It plots f and F
mesh(z(1:100:2500), T(1:100:1250), f(1:100:1250,1:100:2500));
mesh(z(1:100:2500), T(1:100:1250), F(1:100:1250,1:100:2500));