Nelle grotte francesi di una località chiamata Lascaux degli uomini dipinsero il mondo che li circondava, più di diecimila anni fa. E’ tutto ancora lì, come lo hanno lasciato.
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 [1] e la cui prevalenza nella popolazione generale è superiore a quella della sclerosi multipla [2]. 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 [3]. Di questi, l’80% non è in grado di svolgere una attività lavorativa [4] e il 25% è costretto in casa o a letto dalla severità dei sintomi [5]. Il loro funzionamento fisico e mentale sarà compromesso per sempre – solo il 5% dei pazienti guarisce [6] – 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 [7]. 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.
Paolo Maccallini
Bibliografia
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
Agenas
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.
Terapia cognitivo-comportamentale
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 terapia cognitivo comportamentale (CBT) (un tipo di psicoterapia) 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.
Opere citate
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).
Figure 1. Density (left) and cumulative distribution function (right) for the maximum of the m components of a normal random vector. Numerical integration with MATLAB (by Paolo Maccallini).
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).
Figure 2. The time in which Z reaches its largest value (left) and the largest value (right) in the function of the dimension of the normal random vector. The two curves in dotted lines are obtained through numerical integration (Simpson’s method) while the continuous lines are the function obtained by Majorana in an analytical way.
% file name = massimo_vettore_normale
% date of creation = 22/05/2019
clear all
delta = 0.01;
n(1) = 0.
for i=2:1:301;
n(i) = delta + n(i-1);
n_2(i) = - n(i);
end
for i=1:1:301
f(i) = 0.39894228*( e^( (-0.5)*( n(i)^2 ) ) );
end
for i=1:1:3
sigma(1) = 0.;
sigma(3) = sigma(1) + delta*( f(1) + ( 4*f(2) ) + f(3) )/3;
sigma(2) = sigma(3)*0.5;
for j=2:1:299
sigma(j+2) = sigma(j) + delta*( f(j) + ( 4*f(j+1) ) + f(j+2) )/3;
end
end
for i=1:1:301
F(i) = 0.5 + sigma(i);
F_2(i) = 1-F(i);
end
for i=1:1:100;
m(i) = i;
end
for i=1:1:301
for j=1:1:100
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 ) ) );
endfor
endfor
figure (1)
mesh(m(1:2:100),n(1:10:301),F_Z(1:10:301,1:2:100));
grid on
hold on
mesh(m(1:2:100),n_2(2:10:301),F_Z_2(2:10:301,1:2:100));
xlabel('m');
ylabel('t');
legend('F',"location","NORTHEAST");
figure (2)
mesh(m(1:2:100),n(1:10:301),f_Z(1:10:301,1:2:100));
grid on
hold on
mesh(m(1:2:100),n_2(2:10:301),f_Z_2(2:10:301,1:2:100));
xlabel('m');
ylabel('t');
legend('f',"location","NORTHEAST");
Asymptotic series
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. Note: the trascendental equation that has been solved here seems the one whose solution is the Lambert W function, but it is not the same!
Figure 3. The value of x in which the density of Z reaches its largest value. Majorana’s solution (continuous line) and a numerical solution obtained by means of Newton’s method (dotted line).Figure 4. From the original manuscript by Majorana, which can be found here (page 70). These are some of the passages that I have discussed here.
% file name = tangenti
% date of creation = 08/06/2019
clear all
x(1) = 1; %the initial guess
for i=1:1:100
m(i) = i;
end
for i=1:1:100
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 )
break;
endif
max_t (i) = x(j);
endfor
endfor
% the aproximations by Majorana
for j=1:1:100
max_t_M (j) = sqrt(log(j^2)) - ( log(sqrt(2*pi*log(j^2)))/sqrt(log(j^2)) );
endfor
% it plots the diagrams
plot(m(1:1:100),max_t (1:1:100),'.k','Linewidth', 1)
xlabel('m')
ylabel('time for maximum value')
grid on
hold on
plot(m(1:1:100),max_t_M (1:1:100),'-k','Linewidth', 1)
legend('numerical integration',"Majorana's approximation", "location", 'southeast')
Epilogue
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 (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 have been 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 (paragraphs 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 7 ME/CFS patients. In the last paragraph, I propose an alternative definition of CCI, with the introduction of Euler’s angles.
Figure 1. Left. The midline slice of the set of sagittal sections of an MR study of the brain of the author of this article. In the lower part of the image, we can see the section of the axis, with the odontoid process wedged between the anterior arch of the atlas and the ventral layer of the dura. The posterior arch of the atlas can also be seen just below the posterior edge of the foramen magnum. Above the foramen magnum, the brainstem with its three components: the medulla (or medulla oblongata), pons, and midbrain. Right. Lateral-posterior view of the atlas and the axis. They make up a one degree of freedom kinematic pair with the rotation axis corresponding to the axis of the odontoid. By Paolo Maccallini.
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).
3. Points
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.
Figure 2. This is the median slice from the sagittal sections of a T1-weighted magnetic resonance of the head. In blue, the three points used for the measure of the Grabb’s line and of the clival-canal angle. In red, the definitions of rostral, caudal, ventral and dorsal. By Paolo Maccallini.
4. Lines
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.
Figure 3. On this middle sagittal section, the points CP, B, vCMD and PIC2 have been reported. Since the sphenooccipital suture is not clearly visible, the point CP has been identified measuring 2 centimetres from B, along the dorsal cortical bone. The lines CS, PAL, BC2L and GL have then been identified. The Grabb’s measure is of 0.8 centimetres while the clival-canal angle is 142°. By Paolo Maccallini.
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.
Figure 4. Left. ME/CFS patient, female. The clival-canal angle is 146° while the Grabb’s measure is 0.8 cm. Right. ME/CFS patient, male. The clival-canal angle is 142° while the Grabb’s measure is about 0.6 cm. By Paolo Maccallini.
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).
Figure 5. The Horizontal Harris measure in the sagittal section of the MRI of a patient. Highlighted by the red arrow. By Paolo Maccallini.
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.
Figure 6. Chamberlain’s line (in red). The distance between the Chamberlain’s line and the closest point of the tip of the dens is – 4 mm, in this subject. The minus before the number indicates that the selected point of the dens is below the line. By Paolo Maccallini.
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).
Figure 7. McRae’s line (in green). The distance between the McRae’s line and the closest point of the tip of the dens is – 6 mm, in this subject. The minus before the number indicates that the selected point of the dens is below the line. By Paolo Maccallini.
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).
Figure 8. In this model, it is assumed that CCI may produce a clinical picture similar to ME/CFS because in both the conditions there is a pathology of the brainstem. By Paolo Maccallini.
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.
Table 1. GM: Grabb’s measure; HHM: horizontal Harris measure; BDI: basion-dens interval; CDD: distance between dens and Chamberlain’s line; MDD: distance between dens and McRae’s line; CXA: clival-canal angle; POTS: postural orthostatic tachycardia syndrome; PEM: post-exertional malaise; μ: mean value; σ: standard deviation. By Paolo Maccallini.
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.
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.
Figure 7. The orientation of (Ω; ξ, η, ζ) with respect to (O; x, y, z) is univocally determined by the angles ψ, φ, θ. N is the line of nodes, defined as the intersection between plane xy and plane ξη. By Paolo Maccallini.
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.
