Nei giorni scorsi mi è stato chiesto di scrivere un breve discorso da pronunciare davanti al ministro Giulia Grillo, in una delle tante occasioni in cui la aristocrazia contemporanea concede la voce alle istanze degli elettori. Non ho mai considerato i politici degli interlocutori interessanti, per due ordini di ragioni, collegate fra loro: la prima è che non li ritengo artefici della storia, la seconda è che, in media e con qualche proverbiale eccezione a suffragio della regola, si tratta di persone mediocri.
Del resto una persona in gamba di certo non si dedica alla politica, non ne avrebbe il tempo né la motivazione. Immaginate se Fancis Crick, anziché prendere d’assalto la struttura del DNA a colpi di funzioni di Bessel, si fosse dedicato alla amministrazione delle piccolezze pubbliche: che perdita, che delitto contro l’umanità! Euclide, che da solo ha prodotto il volume che è secondo per diffusione solo alla Bibbia (che però vanta decine di autori tra evangelisti e profeti, oltre il Creatore dell’Universo), ci avrebbe lasciati orfani della spina dorsale della nostra formazione se, anziché donarsi alla matematica, si fosse fatto traviare dall’agone politico. Se Newton si fosse perso in dibattiti sulla cosa pubblica, la prima equazione differenziale della storia avrebbe dovuto attendere forse decenni, ma non sarebbe mai stata così bella. I ministri passano senza lasciare traccia, le persone grandi – magari travestite da miserabili (si pensi a Van Gogh o a Srinivasa Ramanujan) – cambiano il mondo per sempre e sempre per il meglio.
Insomma, gli individui di talento non devono occuparsi di politica, è umiliante. E altrettanto umiliante è, a mio avviso, cercare di interloquire con gli amministratori della polis. Ciò nonostante, poiché mi è stata fatta questa richiesta accorata, ho scritto quanto segue, controvoglia e consapevole di aver compiuto un passo ulteriore nel mio personale cammino verso l’inferno.
Illustre Ministro e gentili convenuti,
attraverso queste poche righe apprenderete della esistenza di una patologia a cui è associato un livello di disabilità non inferiore a quello della sclerosi multipla, dell’artrite reumatoide o dell’insufficienza renale  e la cui prevalenza nella popolazione generale è superiore a quella della sclerosi multipla . Di questa malattia non avete probabilmente mai sentito parlare ma è possibile che ciascuno di voi abbia conosciuto almeno una volta nella vita una persona che ne è afflitta: un’amica, o il figlio di un collega; individui produttivi fino a un certo momento della loro esistenza, poi inspiegabilmente scomparsi dalla scuola o dal lavoro, per un male che non riescono a chiamare per nome.
Si stima che 240 mila italiani ne soffrano, circa lo 0.4% della popolazione . Di questi, l’80% non è in grado di svolgere una attività lavorativa  e il 25% è costretto in casa o a letto dalla severità dei sintomi . Il loro funzionamento fisico e mentale sarà compromesso per sempre – solo il 5% dei pazienti guarisce  – e la loro vita sarà ridotta in molti casi a una sopravvivenza improduttiva.
Riuscite a ricordare la vostra peggiore influenza? Una fatica prepotente vi costringe a letto, la mente diventa incapace di formulare pensieri, è necessaria assistenza anche per piccoli gesti quotidiani. Ecco, in prima approssimazione è possibile affermare che le persone affette da questa condizione sperimentino quel tipo di compromissione tutti i giorni, dal momento dell’esordio della patologia. E la caratteristica dei pazienti, il sintomo patognomonico della condizione, è che qualunque tentativo di evadere dalla cattività fisica e mentale peggiora i sintomi. Ogni sforzo, anche il più triviale, acuisce la patologia. L’età media di insorgenza della malattia è 33 anni, ma sono documentati casi di esordio a meno di 10 anni di vita e a più di 70 . E naturalmente, più precoce è l’esordio e maggiori sono i danni nella vita del paziente: i ragazzi perderanno l’istruzione, lo sport, gli amici e il futuro; gli adulti dovranno rinunciare al lavoro e alla famiglia.
Chiunque di voi o dei vostri congiunti può sviluppare la malattia domani. In quel caso malaugurato, al termine di un percorso annoso tra vari ospedali e specialisti, dopo aver speso i risparmi in cerca di una risposta, scoprireste che nessuna cura potrà restituirvi la salute. Non conta quanto talentuosi foste prima, quante risorse avete a disposizione, non conta la vostra posizione sociale: la vita sarà rovinata per sempre.
Tutti coloro che professionalmente si occupano di questi pazienti, dai ricercatori ai medici, si riferiscono alla patologia con la sigla ME/CFS, un nome con una lunga storia, troppo lunga da raccontare qui.
Non spetta a me parlare delle anomalie immunitarie, metaboliche e neurologiche documentate in questi pazienti negli ultimi 30 anni, ma fornirò al Ministro, e a chiunque sia interessato, la documentazione scientifica raccolta sin qui sulla ME/CFS, tra cui in particolare una revisione della letteratura ad opera della prestigiosa National Academy of Medicine, la quale nel 2015 ha definito la ME/CFS “una malattia multisistemica, seria, cronica che limita drammaticamente la vita di chi ne è colpito” (7).
Ad oggi non è possibile salvare queste persone, ma c’è una comorbidità che le affligge su cui si può e si deve intervenire: la cronica mancanza di fondi per la ricerca e di assistenza sanitaria ed economica. C’è bisogno di ambulatori dedicati, di consapevolezza diffusa e di ricercatori che indirizzino i loro sforzi verso questo problema. In Italia abbiamo tutta la tecnologia e le competenze scientifiche per giocare un ruolo di primo piano nella corsa alla ricerca di una cura, ricerca che attualmente vede impegnati alcuni gruppi sparsi nel pianeta, con risorse umane ed economiche tragicamente troppo modeste. Con una sua decisione, Ministro, si può cambiare il corso di queste vite lasciate fino ad oggi sole ad affrontare lo iato muto della loro esistenza.
Falk Hvidberg, M, et al. The Health-Related Quality of Life for Patients with Myalgic Encephalomyelitis / Chronic Fatigue Syndrome (ME/CFS). PLoS One. . 6 Jul 2015, Vol. 10, 7.
Jason, LA, et al. Differentiating Multiple Sclerosis from Myalgic Encephalomyelitis and Chronic Fatigue Syndrome. Insights Biomed. 12 Jun 2018, Vol. 2, 11.
Jason, LA, et al. A community-based study of chronic fatigue syndrome. Arch Intern Med. 11 Oct 1999, Vol. 159, 18, p. 2129-37.
Klimas, N e Patarca-Montero, R.Disability and Chronic Fatigue Syndrome: Clinical, legal, and patient perspectives. Binghamton : Routledge, 1998. p. 124.
Pendergrast, T, et al. Housebound versus nonhousebound patients with myalgic encephalomyelitis and chronic fatigue syndrome. Chronic Illn. Dec 2016, Vol. 12, 4, p. 292-307.
Cairns, R e Hotopf, M. A systematic review describing the prognosis of chronic fatigue syndrome. Occup Med (Lond). . Jan 2005, Vol. 55, 1, p. 20-31.
Beyond Myalgic Encephalomyelitis/Chronic Fatigue Syndrome: Redefining an Illness. Institute of Medicine. Washington (DC) : National Academies Press (US), 2015.
Nel 2014, un gruppo di esponenti del mondo biomedico e associativo italiano, riuniti dalla Agenzia Nazionale per i Servizi Sanitari Regionali (Agenas), ha prodotto un volume sulla Sindrome da Fatica Cronica (CFS) (R) i cui contenuti includono uno studio epidemiologico della patologia in Italia basato sulla analisi delle schede di dimissione ospedaliera (SDO) tra il 2001 e il 2010 e una revisione della letteratura scientifica internazionale sulla patologia. Gli scopi del documento sono quelli di educare medici, pazienti e loro familiari sulla CFS. Questo lavoro presenta scopi e metodi simili a quelli di un lavoro del 2015 che ha visto impegnati negli Stati Uniti un gruppo di esperti riuniti dalla prestigiosa Academy of Medicine (già Institute of Medicine) (R), con la differenza che in quest’ultimo caso il processo di revisione della letteratura ha anche partorito un nuovo criterio diagnostico per la patologia.
In 224 pagine, divise in 14 capitoli, sono affrontate non solo le anomalie genetiche, immunitarie, neuroendocrinologie e cognitive di questa popolazione di pazienti, ma sono riportati anche dati inediti sulla prevalenza della patologia nel nostro paese; non prima di avere fornito una panoramica sui diversi criteri diagnostici disponibili e senza trascurare i possibili interventi terapeutici. Tuttavia, dal confronto del documento Agenas con il volume della Academy of Medicine emergono almeno due differenze (vedi tabella) che li pongono, a mio modesto parere, in contraddizione fra loro.
Academy of Medicine
Fattori psicologici e/o psichiatrici
La CFS è una condizione medica, non è una malattia psichiatrica né psicologica.
La componente somatica e quella psicologica hanno lo stesso peso nella genesi dei sintomi.
I disturbi cognitivi nei pazienti con CFS/ME sembrano essere collegati con disagi di natura psicologica, specie nel sesso femminile.
Differenze nelle metodologie, nelle misure dei risultati, nei criteri di selezione dei soggetti e altri fattori rendono difficile trarre conclusioni circa l’efficacia di questi interventi.
Discreto successo per aumentare l’attività dei pazienti.
Esercizio aerobico graduale
Questo tipo di intervento è efficace nelle donne affette da CFS/ME.
Si osserva infatti che se gli esperti d’oltreoceano sanciscono fin dall’abstract che la CFS “è una condizione medica, non è una malattia psichiatrica né psicologica”, il documento nostrano dedica il capitolo 12 alle comorbità psichiatriche nei pazienti CFS e conclude che in questa patologia “componenti somatiche e aspetti psicologici si embricano in maniera complessa”, volendo con questa espressione ricercata significare che le due componenti menzionate hanno pari peso nella eziologia dei sintomi. Gli Autori italiani incoraggiano a non trascurare l’ambito psicologico perché “Escludere una delle due componenti, se può portare dei vantaggi a breve termine, a lungo termine rischia di privare il paziente di un trattamento personalizzato ed integrato” (pag. 189). In particolare, il documento Agenas non esclude un fattore causale della componente psicologica sui deficit cognitivi affermando che “I disturbi cognitivi nei pazienti con CFS/ME sembrano essere collegati con disagi di natura psicologica, specie nel sesso femminile” (Cap. 8, pag. 115).
Altra asimmetria fra i due documenti si ravvisa nelle raccomandazioni sui trattamenti. Il documento Agenas apre il capitolo sui trattamenti riconoscendo il valore terapeutico della psicoterapia cognitivo comportamentale (CBT) e dell’esercizio aerobico graduale (graded exercise therapy, GET) (cap. 12). Gli Autori stranieri, dal canto loro, concludono in Appendice C che “I lavori di Taylor e Kielhofner (2005), coerentemente con le conclusioni della revisione sistematica di Ross e colleghi (2002, 2004), non hanno fornito alcuna prova per quanto riguarda l’efficacia riabilitativa della CBT e/o della GET. Differenze nelle metodologie, nelle misure dei risultati, nei criteri di selezione dei soggetti e altri fattori rendono difficile trarre conclusioni circa l’efficacia di questi interventi”.
In tabella sono riassunte le contraddizioni rilevate fra i due documenti.
A.V.Determinanti della salute della donna, medicina preventiva e qualità delle cure: Chronic Fatigue Syndrome “CFS”. Roma : Age.na.s., 2014.
Beyond Myalgic Encephalomyelitis/Chronic Fatigue Syndrome: Redefining an Illness. Institute of Medicine. Washington (DC) : National Academies Press (US), 2015.
In questo frammento (vedi sotto) della sessione domande/risposte dopo la proiezione del documentario Unrest a Torino, si parla dei disturbi cognitivi nella ME/CFS. Come introduzione a questo argomento trovo pertinente una osservazione del neurologo Kristian Sommerfelt della università di Bergen (Norvegia):
“Questo [il distrubo cognitivo] è un sintomo tipico della ME e quello che secondo me causa le maggiori limitazioni. Io non credo che le limitazioni più importanti siano imputabili al fatto che i pazienti sperimentano fatica a seguito di attività fisiche o anche semplicemente quando devono stare seduti. Se fosse solo quella la difficoltà, credo che numerosi pazienti avrebbero avuto una vita molto migliore. No, il problema è che solo tentare di usare il proprio cervello, porta alla incapacità di utilizzarlo. La mente rallenta oppure – in alcuni casi – si blocca del tutto; dipende dal livello di gravità. (R)”
E’ utile ricordare che la diagnosi di ME/CFS non richiede necessariamente la presenza di deficit cognitivi per essere fatta. Tuttavia secondo gli ultimi criteri (in ordine cronologico) nel caso in cui il paziente non lamenti disturbi cognitivi, deve però soffrire di intolleranza ortostatica (ovvero di POTS o di ipotensione ortostatica) (IOM, 2015). E siccome nella intolleranza ortostatica sono descritti disturbi cognitivi, ne segue che implicitamente questi deficit sono necessari alla diagnosi. Tuttavia, anche se presenti, possono avere severità e caratteristiche molto diverse da paziente a paziente. Dal mio osservatorio di paziente curioso, ho notato che molti soggetti con diagnosi di ME/CFS non lamentano né disturbi cognitivi né intolleranza ortostatica. E la mia idea è che la patologia clinicamente definita dai criteri IOM 2015 sia in realtà un sottoinsieme relativamente raro in seno al gruppo definito dai criteri Fukuda del 1994.
Sono andato a Torino con lo scopo principale di riuscire a parlare di questo aspetto, prima che di ogni altra cosa. Quotidianamente vivo non solo la mia frustrazione dovuta a una mente non funzionante da quasi 20 anni, ma anche la sofferenza lancinante di alcuni pazienti giovanissimi con cui sono in contatto, che patiscono in silenzio l’esclusione dalle proprie vite a causa di questo problema. Trovo doloroso anche solo riguardare il video, perché nel quotidiano spesso cerco di sfuggire alla analisi lucida e impietosa che ho fatto in questa occasione. Ma spero che sia utile, che serva.
I disturbi cognitivi più frequentemente riportati in questa popolazione consistono in un rallentamento della velocità con cui la mente processa le informazioni. Mi sono reso conto qualche settimana fa che è possibile dimostrare con semplici passaggi (usando una rete che modellizzi nuclei di materia grigia collegati da materia bianca) che questo tipo di deficit si evidenzia soprattutto nelle attività mentali che richiedono la collaborazione di più aree cerebrali: cioè le attività più complesse. Per altro, se questo fosse vero, si spiegherebbe perché questi deficit non vengono rilevati nei test cognitivi usuali, i quali misurano l’efficienza delle singole funzioni mentali, e non la loro collaborazione in attività complesse che costituiscono però spesso il centro della nostra vita. Proverò a scrivere la dimostrazione quando starò meglio.
Di seguito due miei disegni che rappresentano – facendo ricorso all’allegoria dell’androide – proprio i disturbi cognitivi.
When Ettore Majorana first met Enrico Fermi, between the end of 1927 and the beginning of 1928, Fermi – who was already an acclaimed scientist in the field of nuclear physics – had just solved an ordinary differential equation of the second order (whose solution is now commonly named the Thomas-Fermi function) – by numerical integration. It required a week of assiduous work for him to accomplish this task, with the aid of a hand calculator. Fermi showed the results (a table with several numbers) to Majorana, who was a 21 years old student of electrical engineering who had some vague idea of switching from the field of boring ways of providing electrical energy for boring human activities, to the quest for the intimate structure of matter, under the guide of Fermi, the brightest Italian scientific star of that period.
Majorana looked at the numerical table, as I said, and said nothing. After two days he came back to Fermi’s lab and compared his own results with the table made by Fermi: he concluded that Fermi didn’t make any mistake, and he decided that it could be worth working with him, so he switched from engineering to physics (Segrè E. 1995, page 69-70).
Only recently it has been possible to clarify what kind of approach to the equation Majorana had in those hours. It is worth mentioning that he not only solved the equation numerically, I guess in the same way Fermi did but without a hand calculator and in less than half the time; he also solved the equation in a semianalytic way, with a method that has the potential to be generalized to a whole family of differential equations and that has been published only 75 years later (Esposito S. 2002). This mathematical discovery has been possible only because the notes that Majorana wrote in those two days have been found and studied by Salvatore Esposito, with the help of other physicists.
I won’t mention here the merits that Majorana has in theoretical physics, mainly because I am very very far from understanding even a bit of his work. But as Erasmo Recami wrote in his biography of Majorana (R), a paper published by Majorana in 1932 about the relativistic theory of particles with arbitrary spin (Majorana E. 1932) contained a mathematical discovery that has been made independently in a series of papers by Russian mathematicians only in the years between 1948 and 1958, while the application to physics of that method – described by Majorana in 1932 – has been recognized only years later. The fame of Majorana has been constantly growing for the last decades.
The notes that Majorana took between 1927 and 1932 (in his early twenties) have been studied and published only in 2002 (Esposito S. et al. 2003). These are the notes in which the solution of the above-mentioned differential equation has been discovered, by the way. In these 500 pages, there are several brilliant calculations that span from electrical engineering to statistics, from advanced mathematical methods for physics to, of course, theoretical physics. In what follows I will go through what is probably the less difficult and important page among them, the one where Majorana presents an approximated expression for the maximum value of the largest of the components of a normal random vector. I have already written in this blog some notes about the multivariate normal distribution (R). But how can we find the maximum component of such a vector and how does it behave? Let’s assume that each component has a mean of zero and a standard deviation of one. Then we easily find that the analytical expressions of the cumulative distribution function and of the density of the largest component (let’s say Z) of an m-dimensional random vector are
We can’t have an analytical expression for the integral, but it is relatively easy to use Simpson’s method (see the code at the end of this paragraph) to integrate these expressions and to plot their surfaces (figure 1).
Now, what about the maximum reached by the density of the largest among the m components? It is easy, again, using our code, to plot both the maximum and the time in which the maximum is reached, in function of m (figure 2, dotted lines). I have spent probably half an hour in writing the code that gives these results, but we usually forget how fortunate we are in having powerful computers on our desks. We forget that there was a time in which having an analytical solution was almost the only way to get a mathematical work done. Now we will see how Majorana obtained the two functions in figure 2 (continuous line), in just a few passages (a few in his notes, much more in mine).
% file name = massimo_vettore_normale
% date of creation = 22/05/2019
delta = 0.01;
n(1) = 0.
n(i) = delta + n(i-1);
n_2(i) = - n(i);
f(i) = 0.39894228*( e^( (-0.5)*( n(i)^2 ) ) );
sigma(1) = 0.;
sigma(3) = sigma(1) + delta*( f(1) + ( 4*f(2) ) + f(3) )/3;
sigma(2) = sigma(3)*0.5;
sigma(j+2) = sigma(j) + delta*( f(j) + ( 4*f(j+1) ) + f(j+2) )/3;
F(i) = 0.5 + sigma(i);
F_2(i) = 1-F(i);
m(i) = i;
F_Z (i,j) = F(i)^j;
F_Z_2 (i,j) = F_2(i)^j;
f_Z (i,j) = 0.39894228*j*( F(i)^(j-1) )*( e^( (-0.5)*( n(i)^2 ) ) );
f_Z_2 (i,j) = 0.39894228*j*( F_2(i)^(j-1) )*( e^( (-0.5)*( n(i)^2 ) ) );
I have always been fascinated by integrals since I encountered them a lifetime ago. I can still remember the first time I learned the rule of integration by parts. I was caring for my mother who was dying. That night I was in the hospital with her, but she couldn’t feel my presence, she had a tumour in her brain and she was deteriorating. And yet I was not alone, because I had my book of mathematics and several problems to solve. But when my mind was hit by the disease for the first time, about a year later, and I lost the ability to solve problems, then real loneliness knocked at my door.
Now, why am I talking about the integration by parts? Well, I have discovered a few days ago, while studying Majorana’s notes, that integration by parts – well known by students to be a path towards recursive integrations that usually leads to nowhere – is in fact a method that can be useful for developing series that approximate a function for large values of x (remember that Taylor’s polynomials can approximate a function only for values of x that are close to a finite value x_0, so we can’t use them when x goes to ∞). Majorana used one such a series for the error function. He developed a general method, which I tried to understand for some time, without being able to actually get what he was talking about. His reasoning remained in the back of my mind for days, while I moved from Rome to Turin, where I delivered a speech about a paper on the measure of electric impedance in the blood of ME/CFS patients; and when I cried, some minutes later, looking at my drawings put on the screen of a cinema, Majorana was with me, with his silence trapped behind dark eyes. A couple of days later, I moved to a conference in London, searching for a cure that could perhaps allow my brain to be normal again and I talked with a huge scientist that once worked with James Watson. Majorana was there too, in that beautiful room (just a few metres from Parliament Square), sitting next to me. I could feel his disappointment, I knew that he would have found a cure, had he had the chance to examine that problem. Because as Fermi once said to Bruno Pontecorvo, “If a problem has been proposed, no one in the world can resolve it better than Majorana” (Esposito S. et al. 2003). Back in Rome, I gave up with the general method by Majorana and I found the way to calculate the series from another book. The first tip is to write the error function as follows:
Now by integrating by parts, one gets
But we can integrate by parts one other time, and we get
And we can go on and on with integration by parts. This algorithm leads to the series
whose main property is that the last addend is always smaller (in absolute value) than the previous one. And even though this series does not converge (it can be easily seen considering that the absolute value of its generic addend does not go to zero for k that goes to ∞, so the Cauchy’s criteria for convergence is not satisfied) it gives a good approximation for the error function. From this series, it is easy to calculate a series for the Gaussian function (which is what we are interested in):
A clever way to solve a transcendental equation if you don’t want to disturb Newton
Taking only the first two terms of the series, we have for the cumulative distribution function of Z the expression:
The further approximation on the right is interesting, I think that it comes from a well-known limit:
Now we can easily calculate the density of Z by deriving the cumulative distribution function:
With a further obvious approximation, we get:
In order to find the value of x in which this density reaches its largest value, we have to search for the value of x in which its derivative is zero. So we have to solve the following equation:
Which means that we have to solve the transcendental equation:
Majorana truncated the second member of the equation on the right and proposed as a solution the following one:
Then he substituted again this solution in the equation, in order to find ε:
With some further approximations, we have
So Majorana’s expression for the value of x in which the density of Z reaches its maximum value is
I have tried to solve the transcendental equation with Newton’s method (see the code below) and I found that Majorana’s solution is a very good one (as you can see from figure 3). Now, If we compare the approximation by Majorana with what I obtained using numerical integration at the beginning (figure 2) we see that Majorana found a very good solution, particularly for the value of x_M.
% file name = tangenti
% date of creation = 08/06/2019
x(1) = 1; %the initial guess
m(i) = i;
for j = 2:1:1000
f(j-1) = exp( 0.5*( x(j-1)^2 ) ) - ( m(i)/( x(j-1)*sqrt(2*pi) ) );
f_p(j-1) = x(j-1)*exp( 0.5*( x(j-1)^2 ) ) + ( m(i)/( (x(j-1)^2)*sqrt(2*pi) ) );
x(j) = x(j-1) - ( f(j-1)/f_p(j-1) );
if ( abs(x(j)) < 0.001 )
max_t (i) = x(j);
% the aproximations by Majorana
max_t_M (j) = sqrt(log(j^2)) - ( log(sqrt(2*pi*log(j^2)))/sqrt(log(j^2)) );
% it plots the diagrams
plot(m(1:1:100),max_t (1:1:100),'.k','Linewidth', 1)
ylabel('time for maximum value')
plot(m(1:1:100),max_t_M (1:1:100),'-k','Linewidth', 1)
legend('numerical integration',"Majorana's approximation", "location", 'southeast')
From 1934 to 1938 Majorana continued his studies in a variety of different fields (from game theory to biology, from economy to quantistic electrodynamics), but he never published again (R), with the exception for a work on the symmetric theory of electrons and anti-electrons (Majorana E. 1937). But it has been concluded by biographers that the discoveries behind that work were made by Majorana about five years earlier and yet never shared with the scientific community until 1937 (Esposito S. et al. 2003). And in a spring day of the year 1938, while Mussolini was trying his best to impress the world with his facial expressions, Ettore became a subatomic particle: his coordinates in space and their derivatives with respect to time became indeterminate. Whether he had lived in a monastery in the south of Italy or he had helped the state of Uruguay in building its first nuclear reactor; whether he had seen the boundless landscapes of Argentina or the frozen depth of the abyss, I hope that he could have found, at last, what he was so desperately searching for.
He had given his contribution to humanity, so whatever his choice has been, his soul was already safe. And as I try to save my own soul, going back and forth from mathematics to biology, in order to find a cure, I can feel his presence. The eloquence of his silence trapped behind dark eyes can always be clearly heard if we put aside the noise of the outside world. And it tells us that Nature has a beautiful but elusive mathematical structure which can nevertheless be understood if we try very hard.
In the meanwhile, I write these short stories, like a mathematical proof of my own existence, in case I don’t have further chances to use my brain.
Sabato scorso ho avuto il piacere di partecipare alla prima proiezione italiana del documentario Unrest, un pregevole film che racconta il viaggio di Jennifer Brea, dalla salute alla malattia. La proiezione è stata organizzata da Caterina Zingale, una donna minuta con le determinazione di un gigante, e dalle due maggiori associazioni CFS della penisola, la Associazione CFS Onlus e la CFSME Associazione Italiana.
Alcuni giorni prima della manifestazione mi sono deciso, non senza dubbi e ripensamenti, a parlare dello studio sulla misura della impedenza elettrica nel sangue dei pazienti ME/CFS, una ricerca che potenzialmente può cambiare lo scenario di questa patologia. Non è un argomento semplice, e non c’è un modo semplice per raccontarlo, a meno di non voler rinunciare a una comprensione reale del risultato sperimentale. Non so se sono riuscito nei miei intenti, lo lascio alla vostra considerazione, il video è riportato a seguire. Le slide che ho usato per la presentazione possono essere scaricate qui. Una versione leggermente più lunga delle slide (con informazioni aggiuntive) è disponibile qui. Faccio notare che, poiché le slide contengono delle animazioni, l’unico modo per vederle è scaricarle e poi avviare la presentazione. Un mio articolo su questo argomento è disponibile qui.
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. 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 found some convincing arguments in favour of that link. So here I am, with this new blog post. In paragraph 2 I will 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 and 6). In paragraph 7, I discuss the possible link between craniocervical instability and ME/CFS. Enjoy.
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 mesure and the clival-canal angle.
Clival slope (CS). It connects CP to vCMD. It is also called the Wackenheim Clivus Baseline (Smoker WRK 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 (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).
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). 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, while the CXA and the Grabb’s measure for three ME/CFS patients (the one in figure 3 and the two in figure 4) are collected in the table below (OI stands for orthostatic intolerance).
7. Craniocervical instability and ME/CFS
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).
There are some clues that can potentially link CCI to ME/CFS, as mentioned in the introduction. My interest in this topic aroused some weeks ago because of anecdotal reports of diagnosis of CCI (with subsequent successful surgery, apparently) among ME/CFS patients (R, R), in the absence of formal studies (to my knowledge, at least). And yet there is a substantial overlap between Ehlers-Danlos syndrome hypermobile type (EDS-HT) and ME/CFS, with about 80% of EDS-HT patients meeting the Fukuda criteria (Castori M. et al. 2011) and we know, as mentioned, that CCI is present in Ehlers-Danlos syndrome. Moreover, brainstem abnormalities are well known to be present in ME/CFS, where hypoperfusion (Costa D.c: et al. 1995), hypometabolism (Tirelli U. et al. 1998), reduced volume (Barnden L.R. et al. 2011), microglia activation (Nakatomi Y et al. 2014), and loss of connectivity (Barnden L.R. et al. 2018) in brainstem have been reported. 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. Moreover, the presence of CCI in rheumatoid arthritis might be a clue for a causal role of the immune system in this kind of hypermobility. A link between hypermobility and the immune system has been found also in a condition due to the duplication/triplication of the gene that encodes for tryptase (a proteolytic enzyme of mast cells) (Lyons JJ et al. 2016). CCI can lead to orthostatic intolerance (OI), and OI is widely prevalent in ME/CFS.
So, it is not unreasonable to consider pathology of the craniocervical junction to be involved in some cases of CFS-like symptoms. It might be due to some degree of predisposition to hypermobility and/or to abnormal immune activity. These cases would then be aggravated or triggered by infections, as it is often the case in ME/CFS patients. How to properly classify these patients would be just a matter of nomenclature. But as we all know “A rose by any other name would smell as sweet”.
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 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.
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");