Maximum of a normal random vector

In evidenzaMaximum of a normal random vector

Ettore Majorana hadn’t got MATLAB

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 wrote, 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 least 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

Eq 1.JPG

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).

surfaces.JPG
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 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).

maximum.JPG
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 famous scientist who worked on the human genome project. 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:

Eq 2.JPG

Now by integrating by parts, one gets

Eq 3.JPG

But we can integrate by parts one other time, and we get

Eq 4.JPG

And we can go on and on with integration by parts. This algorithm leads to the series

Eq 5.JPG

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):

Eq 6.JPG

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:

Eq 7.JPG

The further approximation on the right is interesting, I think that it comes from a well-known limit:

Eq 8.JPG

Now we can easily calculate the density of Z by deriving the cumulative distribution function:

Eq 9.JPG

With a further obvious approximation, we get:

Eq 10.JPG

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:

Eq 11.JPG

Which means that we have to solve the transcendental equation:

Eq 12.JPG

Majorana truncated the second member of the equation on the right and proposed as a solution the following one:

Eq 13.JPG

Then he substituted again this solution in the equation, in order to find ε:

Eq 14.JPG

With some further approximations, we have

Eq 15.JPG

So Majorana’s expression for the value of x in which the density of Z reaches its maximum value is

Eq 16.JPG

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 transcendental equation that has been solved here seems the one whose solution is the Lambert W function, but it is not the same!

trascendental.JPG
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).
Vol II.JPG
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 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.

Until time catches me.

 

 

 

On the module of random vectors

In evidenzaOn the module of random vectors

“I have hopes of being able to achieve

something of value through my current studies or

with any new ideas that come in the future.”

J. F. Nash

The bridge over the Arno

In 1999 I was wandering in Pisa with a booklet in the pocket of a worn coat, too short for my frame. That coat was dark blue on the outside, green and red inside, with one mended sleeve and an austere cowl: I was much like a young monk, with his holy book (figure 1). I can remember neither its title nor the author, though. It was an introduction to statistical thermodynamics, with beautiful figures, a coloured cover, and less than 100 pages. It contained the work by Maxwell on the kinetic theory of ideal gasses, along with other material. I borrowed it from the University Library because I was fascinated by the way in which Maxwell was able to describe the properties of gasses with just a few hypotheses and some relatively easy mathematical passages. I felt that there was an enormous attraction in these methods, I realized with pleasure that math could give the power to completely understand and hold in hand physical systems and even, I started speculating, biological ones.

Analisi, coraggio, umiltà, ottimismo.jpg
Figure 1. A small self-portrait I drew in 1999 in an empty page of my pocket-sized French dictionary.

My second favourite composer back then was Gustav Mahler (the favourite one being Basil Poledouris): he represented my own way to classical music and I chose him because he wasn’t among the musicians my father and my brother shared a love for. I felt, during my teens, that I had to find my private space, and I met it one day on a used book stand: a cassette tape of Das Lied von The Erde, with a few sheets containing the translation to Italian of the songs. Mahler was born in 1860, a few weeks after Maxwell published his pivotal work about ideal gasses in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (R) (figure 2). But in 1999 I was repeatedly listening to a CD with a collection of songs sung by Edith Piaf and Charles Trenet, because I was studying French, and I was having a hard time with pronunciation. So, imagine a secular monk in his prime who listens to old French songs while keeping one hand on a book of statistical thermodynamics hidden in his pocket, wherever he goes, wandering in the streets of Pisa, a city which gave birth to Galileo Galilei. This seems a beautiful story, much like a dream, right? Wrong.

cover
Figure 2. Original cover of the journal in which Maxwell published his work on the kinetic theory of ideal gasses.

I had already started my struggle against the mysterious disease that would have completely erased my life in the following years. In the beginning, it had a relapsing-remitting course, so I could sometimes host the hope that I was recovering, only to find myself caught by the evil curse again. At the end of the year 1999, I was losing my mind, I knew that and I was also aware that my holy book couldn’t save me. I clearly remember one evening, I was walking on Ponte di Mezzo, a simple and elegant bridge above the Arno, and I felt that I couldn’t feel sorrow for the loss of my mind: I realized that not only the functions of my brain assigned to rational thinking were gone, but my feelings couldn’t function properly either. In fact, I noted without a true manifestation of desperation that I had lost my emotions. One day, after spending in vain about eleven hours on a single page of a textbook of invertebrate palaeontology, I accepted that I couldn’t read anymore, at least for the moment.

Had I known for sure that I wouldn’t have recovered in the following twenty years, I would have quite certainly taken my own life, jumping from a building; a fate that I have been thinking about almost every day ever since. I considered this possibility during the endless sequence of days in which there has been nothing other than the absence of my thoughts.

The distribution of velocities of an ideal gas and the one hundred years gap

In the already mentioned paper by Maxwell, he derived the probability density of the speed of the molecules of a gas, granted that the three components of the vector of speed are independent random variables (hyp. 1) and that they share the same density (hyp. 2), let’s say f. Moreover, the density of the speed has to be a function only of its module (hyp. 3). These three hypotheses together say that there is a function Φ such that

Cattura.PNG

This is a functional equation (i.e. an equation in which the unknown is a function) whose solution is not detailed in Maxwell’s work. But it can be easily solved moving to polar coordinates (see figure 3) and deriving with respect to θ both members (the second one gives naught since it depends only to the distance from the origin).

space of velocities.png
Figure 3. The space of velocities and its polar coordinates.

Another way to solve the functional equation is to use the method of Lagrange’s multipliers, searching for the extremes of the density of the velocity, when its module is fixed. In either case, we obtain the differential equation:

differential equation.png

which leads to the density for each component of the speed:

density.PNG

where σ can’t be determined using only the three hypotheses mentioned above. Considering then the well-known law of ideal gasses (pV=nRT) and an expression for p derived from the hypothesis that the collisions between the molecules of gas and the container are completely elastic, Maxwell was able to conclude that:

varianza.png

where m is the mass of the molecule of gas, T is the absolute temperature and K_B is the Boltzmann’s constant. It was 1860, Mahler’s mother was going to deliver in Kaliště, Charles Darwin had just released his masterpiece “On the origin of species”, forced to publish much earlier than what he had planned because of the letter he had received from Wallace, in which he described about the same theory Darwin had been working on for the previous 20 years. In the same point in time, Italy was completing its bloody process of unification, with the Mille expedition, led by Giuseppe Garibaldi.

But the functional equation I have mentioned at the beginning of this paragraph has brought with it a mystery for years, until 1976, when an employee at General Motors Corporation published a short note in the American Journal of Physics [R] in which he showed how Maxwell’s functional equation is, in fact, an example of the well known Cauchy’s functional equation:

Cauchy.png

In order to prove that, you just have to consider the following definition:

Nash.png

given that

Nash2.png

The name of the mathematician who made this observation is David H. Nash, and he has the merit of finding something new in one of the most known equation of physics, an equation that is mentioned in every book of thermodynamics, an equation that has been considered by millions of students in more than a century. It was 1976, my mother was pregnant with my brother; Alma, Gustav Mahler’s wife, had died about ten years before.

sphere.PNG
Figure 4. A sphere with a radius of z. The integrals of the density of within this sphere has the obvious meaning of the repartition function of Z.

Module of random vectors

Once Maxwell found the density of probability for each component of the speed of the molecules of an ideal gas, he searched for the density of the module of the speed. There is a relatively simple way of doing that. With the following notation

notazione.png

we have that the repartition function of Z is given by the integrals of the density of X within the sphere in figure 4. We have:

repartition function.PNG

The second expression is the same as above but in polar coordinates. Then we can obtain the density of Z by derivation of the repartition function. And this method can be extended to an m-dimensional space. This was the method used by Maxwell in his paper. And yet, there is another way to obtain the expression of the module of a random vector: I have explored it in the last months, during the rare hours in which I could function. By the way, only in the Summer of 2007 I was able to study the kinetic theory by Maxwell, eight years after I borrowed the holy book. Such a waste.

The hard way towards the density of the module of a random vector

When a student starts studying statistics, she encounters a list of densities: the normal distribution, the gamma distribution, the exponential distribution etc. Then there are several derived distributions that arise when you operate sums, roots extractions etc. on random variables. In particular, if f_X is the densty of X and Y = X², then we have

Quadrato.PNG

On the other hand, if Y = √X we have

Radice.PNG

Another important result that we have to consider is that given

linera combination.PNG

then

linear combination 2.PNG

By using these results I have been able to find that the expression of the density for  the module of an m-dimensional random vector is:

module.PNG

In particular, for m = 3 we have

module 2.PNG

Bessel surface.PNG
Figure 5. A plot of the modified Bessel function.

The case of normal random vectors: the modified Bessel function

In particular, if the random vector has dimension 3 and its components are normal random variables with the same expected value and variance, we have that the density of its module is given by

modulo normale.PNG

where I_0 is the modified Bessel function, which is one solution of the differential equation:

Bessel equation.PNG

whose name is modified Bessel equation. The integral expression of the modified Bessel function is:

Bessel function.PNG

I have coded a script in Matlab which integrates numerically this function (available here for download) which plots the surface in figure 5 and also gives the following table of values for this function.

Bessel table.PNG

The following is the flowchart of the script I coded.

flux chart.png

The case of normal random vectors with a naught expected value: the upper incomplete gamma function

If we consider random variables with an average that is zero (this is the case with the components of speed in ideal gasses), then the density is given by

Chi 3 scaled.PNG

which is a Chi distribution with 3 degrees of freedom, scaled with a scale parameter given by s = 1/σ. In the expression of the repartition function, ϒ is the lower incomplete gamma function, which is defined as follows:

incomplete gamma.PNG

I have written a code for its numerical integration (available here for download), the output of which is in figure 6.

lower incomlete.PNG
Figure 6. Lower incomplete gamma function.
maxwell surface.PNG
Figure 7. Maxwell-Boltzmann distribution for N2 in function of temperature and speed.
maxwell surface 2.PNG
Figure 8. Maxwell-Boltzmann distribution for Helium, Neon, Argon, at room temperature.

Conclusion, twenty years later

The density of the module of the velocity of the molecules of an ideal gas is, in fact, a scaled Chi distribution with 3 degrees of freedom, and it is given by

maxwell.PNG

It can be numerically integrated with the following script I made for Octave/Matlab, which gives the plot in figure 7. Another similar script gives the plot in figure 8. These plots represent the Maxwell-Boltzmann distribution, the centre of the holy book that an unfortunate boy was carrying in his pocket, all alone, some twenty years ago. He could have easily died by his own hand in one of the several thousand days of mental and physical disability that he had to face alone. Instead, he has survived. Had it been only for finding the Maxwell-Boltzmann distribution following another path, it would have been worth it. But he has found much more, including a bright girl, the magnificent next stage of evolution of human beings.

% file name = legge_Maxwell-Boltzmann_2

% date of creation = 22/02/2019

% it plots the density and the distribution function for the

% Maxwell-Boltzmann distribution considered as a function of temperature

% and speed

clear all

% we define some parameters

K_B = 1.381*10^(-23)                                                 % Boltzmann's constant

m = 4.65*10^(-26)                                                     % mass of the molecule N_2

% we define the array of temperature from 0° C to 1250° C

T (1) = 273.15

for i = 2:1:1250

T (i) = T (i-1) + 1.;

endfor

% it defines f_gamma in 3/2

f_gamma = sqrt(pi)/2.

% delta of integration

delta = 1.0

% it defines the array for the abscissa

z (1) = 0.;

for i = 2:1:2500

z (i) = z(i-1)+delta;

endfor

% it defines the density

for j = 1:1:1250

% it defines a constant

c = ( m/(K_B*T(j)) );

for i = 1:1:2500

f (j,i) = ( c^1.5 )*sqrt(2./pi)*( z(i)^2. )*( e^( -0.5*c*z(i)^2. ) );

endfor

% it calculates the ripartition function for He

F (j,1) = 0.;

F (j,3) = F (j,1) + delta*( f(j,1) + ( 4*f(j,2) ) + f(j,3) )/3;

F (j,2) = F (j,3)*0.5;

for k=2:1:2500-2

F (j,k+2) = F (j,k)+delta*( f(j,k)+( 4*f(j,k+1) )+f(j,k+2) )/3;

end

endfor

% It plots f and F

figure (1)

mesh(z(1:100:2500), T(1:100:1250), f(1:100:1250,1:100:2500));

legend('Density',"location","NORTHEAST");

xlabel('speed (m/s)');

ylabel('temperature (Kelvin)');

grid on

figure (2)

mesh(z(1:100:2500), T(1:100:1250), F(1:100:1250,1:100:2500));

legend('Probability',"location","NORTHEAST");

xlabel('speed (m/s)');

ylabel('temperature (Kelvin)');

grid on

 

A complete (preload) failure

A complete (preload) failure

Introduction

Some days ago, David Systrom offered an overview of his work on cardiopulmonary testing in ME/CFS during a virtual meeting hosted by Massachusetts ME/CFS & FM Association and Open Medicine Foundation. In this blog post, I present an introduction to the experimental setting used for Systrom’s work (paragraph 1), a brief presentation of his previous findings (paragraph 2), and an explanation of his more recent discoveries in his cohort of patients (paragraph 3). In paragraph 4 you’ll find a note on how to support this research.

1. Invasive Cardiopulmonary Exercise Testing

It is a test that allows for the determination of pulmonary, cardiac, and metabolic parameters in response to physical exertion of increasing workload. It is, mutatis mutandis, the human equivalent of an engine test stand. A stationary bike with a mechanical resistance that increases by 10 to 50 Watts for minute is usually employed for assessing the patient in a upright position, but a recumbent bike can also be used in some instances. Distinguishing between these two different settings might be of pivotal relevance in ME/CFS and POTS. I shall now briefly describe some of the measurements that can be collected during invasive cardiopulmonary exercise testing (iCPET) and their biological meaning. For a more accurate and in-depth account, please refers to (Maron BA et al. 2013), (Oldham WM et al. 2016). I have used these papers as the main reference for this paragraph, unless otherwise specified.

Gas exchange. A face mask collects the gasses exchanged by the patient during the experiment and allows for monitoring of both oxygen uptake per unit of time (named VO_2) and carbon dioxide output (VCO_2), measured in mL/min. Gas exchange is particularly useful for the determination of the anaerobic threshold (AT), i.e. the point in time at which the diagram of VCO_2 in function of VO_2 displays an abrupt increase in its derivative: at this workload, the patient starts relying more on her anaerobic energy metabolism (glycolysis, for the most part) with a build-up of lactic acid in tissues and blood (see Figure 1).

Figure 1. Diagram of VCO_2 in function of VO_2. The point in which there is a change in the derivative with respect to VO_2 is called “anaerobic threshold” (AT). AT is highlighted with a vertical line in this picture. This diagram is from an actual CPET of a patient.

Oxygen uptake for unit of time at AT (called VO_2max) can be considered an integrated function of patient’s muscular, pulmonary, and cardiac efficiency during exercise. It is abnormal when its value is below 80% of what predicted according to patient’s age, sex, and height. Importantly, according to some studies there might be no difference in VO_2max between ME/CFS patients and healthy controls in this measure unless the exercise test is repeated a day after the first measure: in this case the value maxVO_2 for patients is significantly lower than for controls (VanNess JM et al. 2007), (Snell CR and al. 2013).

Another measure derived from the assessing of gas exchange is minute ventilation (VE, measured in L/min) which represents the total volume of gas expired per minute. The link between VE and VO_2 is as follows:

VO_2\;=\;VE\cdot(inspired\;VO_2\; -\; expired\;VO_2)

Maximum voluntary ventilation (MVV) is the maximum volume of air that is voluntarily expired at rest. During incremental exercise, a healthy person should be able to maintain her VE at a value ∼0.7 MVV and it is assumed that if the ratio VE/MVV is above 0.7, then the patient has a pulmonary mechanical limit during exercise. If VE is normal, then an early AT suggests an inefficient transport of oxygen from the atmosphere to muscles, not due to pulmonary mechanics, thus linked to either pulmonary vascular abnormalities or muscular/mitochondrial abnormalities. It is suggested that an abnormally high derivative of the diagram of VE in function of VCO_2 and/or a high ratio VE/VCO_2 at AT (these are measures of how efficiently the system gets rid of CO_2) are an indicator of poor pulmonary vascular function.

Respiratory exchange ratio (RER) is a measure of the effort that the patient puts into the exercise. It is measured as follows:

RER=\frac{VCO_2}{VO_2}

and an RER>1.05 indicates a sufficient level of effort. in this case the test can be considered valid.

Arterial catheters. A sensor is placed just outside the right ventricle (pulmonary artery, Figure 2) and another one is placed in the radial artery: they allow for measures of intracardiac hemodynamics and arterial blood gas data, respectively. By using this setting, it is possible to indirectly estimate cardiac output (Qt) by using Fick equation:

Qt=\frac{VO_2}{arterial\;O_2 - venous\;O_2}

where the arterial\;O_2 is measured by the radial artery catheter and the venous one is measured by the one in the pulmonary artery (ml/L). An estimation for an individual’s predicted maximum Qt (L/min) can be obtained by dividing her predicted VO_2max by the normal maximum value of  arterial\;O_2 - venous\;O_2 during exercise, which is 149 mL/L:

predicted\; Qt\;max=\frac{predicted\; VO_{2}max}{149 \frac{mL}{L}}

If during iCPET the measured Qt max is below 80% of the predicted maximum cardiac output (as measured above), associated with reduced VO_2max, then a cardiac abnormality might be suspected. Stroke volume (SV), defined as the volume of blood ejected by the left ventricle per beat, can be obtained from the Qt according to the following equation:

Qt=SV\cdot HR\;\xrightarrow\;SV\;=\;\frac{Qt}{HR}\;=\;\frac{\frac{VO_2}{arterial\; O_2 - venous\; O_2}}{HR}

where HR stands for heart rate. One obvious measure from the pulmonary catheter is the mean pulmonary artery pressure (mPAP). The right atrial pressure (RAP) is the blood pressure at the level of the right atrium. Pulmonary capillary wedge pressure (PCWP) is an estimation for the left atrial pressure. It is obtained by the pulmonary catheter. The mean arterial pressure (MAP) is the pressure measured by the radial artery catheter and it is a proxy for the pressure in the left atrium. RAP, mPAP, and PCWP are measured by the pulmonary catheter (the line in red) which from the right atrium goes through the tricuspid valve, enters the right ventricle, and then goes along the initial part of the pulmonary artery (figure 2).

Figure 2. Right atrial pressure (RAP) is the pressure of the right atrium, mean pulmonary arterial pressure (mPAP) is the pressure of the right ventricle, pulmonary capillary wedge pressure (PCWP) is an estimation of the pressure of the left atrium. Mean arterial pressure gives a measure of the pressure of the left ventricle. RAP, mPAP, and PCWP are measured by the pulmonary catheter (the line in red) which from the right atrium goes through the tricuspid valve, enters the right ventricle, and then goes across the initial part of the pulmonary artery (R).

Derived parameters. As seen, Qt (cardiac output) is derived from actual direct measures collected by this experimental setting, by using a simple mathematical model (Fick equation). Another derived parameter is pulmonary vascular resistance (PVR) which is obtained using the particular solution of the Navier-Stokes equations (the dynamic equation for Newtonian fluids) that fits the geometry of a pipe with a circular section. This solution is called the Poiseuille flow, and it states that the difference in pressure between the extremities of a pipe with a circular cross-section A and a length L is given by

\Delta\;P\;=\;\frac{8\pi\mu L}{A^2}Q

where \mu is a mechanical property of the fluid (called dynamic viscosity) and Q is the blood flow (Maccallini P. 2007). As the reader can recognize, this formula has a close resemblance with Ohm’s law, with P analogous to the electric potential, Q analogous to the current, and \frac{8\pi\mu L}{A^2} analogous to the resistance. In the case of PVR, Q is given by Qt while \Delta\;P\;=\;mPAP\;-\;PCWP. Then we have:

PVR\;=\;80\frac{\;mPAP\;-\;PCWP}{Qt}

where the numeric coefficient is due to the fact that PVR is usually measured in \frac{dyne\cdot s}{cm^5} and 1 dyne is 10^5 Newton while 1 mmHg is 1333 N/m².

2. Preload failure

A subset of patients with exercise intolerance presents with preload-dependent limitations to cardiac output. This phenotype is called preload failure  (PLF) and is defined as follows: RAP max < 8 mmHg, Qt and VO_2max <80% predicted, with normal mPAP (<25 mmHg) and normal PVR (<120 \frac{dyne\cdot s}{cm^5}) (Maron BA et al. 2013). This condition seems prevalent in ME/CFS and POTS. Some of these patients have a positive cutaneous biopsy for small-fiber polyneuropathy (SFPN), even though there seems to be no correlation between hemodynamic parameters and the severity of SFPN. Intolerance to exercise in PLF seems to improve after pyridostigmine administration, mainly through potentiation of oxygen extraction in the periphery. A possible explanation for PLF in non-SFPN patients might be a more proximal lesion in the autonomic nervous system (Urbina MF et al. 2018), (Joseph P. et al. 2019). In particular, 72% of PLF patients fits the IOM criteria for ME/CFS and 27% meets the criteria for POTS. Among ME/CFS patients, 44% has a positive skin biopsy for SFPN. One possible cause for damage to the nervous system (both in the periphery and centrally) might be TNF-related apoptosis-inducing ligand (TRAIL) which has been linked to fatigue after radiation therapy; TRAIL increases during iCPET among ME/CFS patients (see video below).

3. Latest updates from David Systrom

During the Massachusetts ME/CFS & FM Association and Open Medicine Foundation Fall 2020 Event on Zoom, David Systrom reported on the results of iCPET in a set of ME/CFS patients. The VO_2max is lower in patients vs controls (figure 3, up). As mentioned before, VO_2max is an index that includes contributions from cardiac performances, pulmonary efficiency, and oxygen extraction rate in the periphery. In other words, a low VO_2max gives us no explanation on why it is low. This finding seems to be due to different reasons in different patients even though the common denominator among all ME/CFS patients of this cohort is a low pressure in the right atrium during upright exercise (low RAP, figure 3, left). But then, if we look at the slope of Qt in function of VO_2 (figure 3, right) we find three different phenotypes. Those with a high slope are defined “high flow” (in red in figure 3). Then we have a group with a normal flow (green) and a group with a low flow (blue). If we look then at the ability to extract oxygen by muscles (figure 3, below) expressed by the ratio

\frac{arterial\;O_2 - venous\;O_2}{HB}

we can see that the high flow patients reach the lowest score. In summary, all ME/CFS patients of this cohort present with poor VO_2max and preload failure. A subgroup, the high flow phenotype, has poor oxygen extraction capacity at the level of skeletal muscles.

Figure 3. The results presented by David Systrom are here displayed around a schematic representation of the circulatory system. VO_2 is a global measure of the efficiency of the circulatory system. CO, which stands for cardiac output (indicated Qt in this blog post) is related to the output of the left half of the heart. RAP is the pressure of the right atrium. By Paolo Maccallini.

Now the problem is: what is the reason for the preload failure? And in the high flow phenotype, why the muscles can’t properly extract oxygen from blood? As mentioned, about 44% of ME/CFS patients in this cohort has SFPN but there is no correlation between the density of small-fibers in the skin biopsies and the hemodynamic parameters. Eleven patients with poor oxygen extraction (high flow) had their muscle biopsy tested for mitochondrial function (figure 4) and all but one presented a reduction in the activity of citrate synthase (fourth column): this is the enzyme that catalyzes the last/first step of Krebs cycle and it is considered a global biomarker for mitochondrial function. Some patients also have defects in one or more steps of the electron transport chain (fifth column) associated with genetic alterations (sixth column). Another problem in high flow patients might be a dysfunctional vasculature at the interface between the vascular system and skeletal muscles (but this might be true for the brain too), rather than poor mitochondrial function.

Figure 4. Eleven patients with high flow (poor oxygen extraction) underwent a muscle biopsy. Mitochondrial function has been assessed in these samples and all the patients but one presented a reduced activity for the enzyme citrate synthase (4th column). Defects in the oxygen transport chain and in the mitochondrial chromosome have also been documented in 4 of them (column 5th and column 6th).

The use of an acetylcholinesterase inhibitor (pyridostigmine) improved the ability to extract oxygen in the high flow group, without improving cardiac output, as measured with a CPET, after one year of continuous use of the drug. This might be due to better regulation of blood flow in the periphery. This paragraph is an overview of the following video:

3. Funding

The trial on the use of pyridostigmine in ME/CFS at the Brigham & Women’s Hospital by Dr. David Systrom is funded by the Open Medicine Foundation (R). This work is extremely important, as you have seen, both for developing diagnostic tools and for finding treatments for specific subgroups of patients. Please, consider a donation to the Open Medicine Foundation to speed up this research. See how to donate.

Six Months

So, it seems that I am improving again. Six months ago I came back from Argentina, where I spent the boreal winter. I felt better there, as I usually do during summer, in Italy. Feeling better means being able to think, to read, to do calculations, to draw. To exist, in one word. And also to move around a bit, which is not truly relevant for me, though.

I came back to Italy at the end of March (blog post), sure that I would have had other months of improvement ahead of me, given that we were at the beginning of Spring. But it hasn’t been the case, I got worse: For six months I haven’t thought, and I have been living horizontally, in silence. There were days in which it seemed that I was starting to improve (like when I recorded this video), but then it didn’t last. I can’t remember these six months, in my subjective time they sum up to a week or less.

Not sure why it happened: perhaps the 48 hours of the chaotic journey back to Italy damaged me so badly that it took half a year for me to regain the status quo ante, or maybe the strange flu I got in March, while in Argentina, made the disease worse. In the life of an ordinary person, this would be a rather exceptional episode, for me it is the rule: the improvements are the rare exception. I have lived like that since I was 20.

And now, because I usually get worse at the end of September, I know that I am about to start my descend to Hell again. And this time I can’t move to the austral hemisphere, because of the pandemic. So what am I supposed to do in the few days of life I have left? I’ll do what I have always wanted to do: applied maths and drawing, with only very short term goals. Something that I can finish.

I share these private vicissitudes only because I think that it is important to let the world know about this struggle. It seems unlikely that I can discover the reason why this curse has stricken my life, but I will continue studying this phenomenon: most of what I study, when I can, is about new tools to apply to my own biology.

The indiscreete rotation

The world must be wonderful, beyond the muffled atmosphere of these rooms and the obstinate curtain of encephalopathy; now that Autumn is still a harmless chrysalis, an apparently unlikely threat, while the industriousness of men swarms again, in search of untouched paths.

The Autumn of intellectual and material adventures, of encounters and discoveries, remains an unfulfilled promise, which I nevertheless do not give up on cultivating. Because I don’t know if Ulysses kissed his stony Ithaca during this season, but I like to think so.

I am perpetually mocked by the indiscreet rotation of the wall clock, which turns on the spot; while Rilke’s panther remains trapped in my chest.

Is it that bad?

Is it that bad?

When I got sick, about 20 years ago, for the first time I started thinking about diseases and loss of health. And I remember coming to the conclusion of how fortunate I was, from a physical standpoint. Not of how fortunate I had been in the past (that was obvious), when I could conquer mountains, running on rocks with my 15 Kg bike on the shoulders; no, of how fortunate I was in that very moment, while confined at home, mostly lying horizontally. I was fortunate, I could still move my hands, I had still all my body, even if I couldn’t use it in the outside world, even if most of the usual activities of a 20 years old man were far beyond reaching, I realized how fortunate I was. And after 20 years I still say that, as for the physical functioning, I am a very lucky man. There are even some years, during the core of the summer, in which I can run for some minutes in the sun. I am blessed.

I am aware that this might offend some patients, but I think that those who have ME/CFS in the vast majority of cases are fortunate too, from a physical standpoint. Yes, it is annoying to need help from others for so many things, but with assistance and some arrangements, you can go on with a productive life… unless you have cognitive impairment.

Actually, I never felt fortunate concerning my cognitive functioning. That has been a true tragedy, I have lost my entire life because of the cognitive damage, by any means because of the limitations of my body. I would have had a meaningful life (according to any standard) even with physical limitations way worse than mine if I had had a functioning brain. But I lost it when I was 20 before I could get the best from it and that is the only real tragedy.

As an example of what I am saying, consider a person like Stephen Hawking: he has been a prolific scientist, he had a stellar career (literally), he shaped our culture with his books for the general public, he was a father of three, and so forth. And yet for most of his life, while he was doing all these things, his physical functioning was way worse than the one of the average ME/CFS patient (worse even of a very severe patient, probably). For many years he could use only the muscles of the head. That’s it. Think about that. Was he missing or invisible? Definitely not!

Was he a disabled man? Yes, technically speaking he was, but in fact, he has never really been disabled, if we judge from his accomplishments. He once said that the disease somehow even helped him in his work, because he could concentrate better on his quantum-relativistic equations¹.

So what I am saying is that if we consider the physical functioning in ME/CFS, it is a negligible problem in most of the cases. The only thing that matters is the impairment in cognition (in the cases in which it is present), especially if it starts at a young age (consider that after you reach your thirties there is very little chance that your brain will give a significant contribution to humanity, even if you are perfectly healthy, so it is not a big loss if you get sick after that age). That is disastrous and there is no wheelchair you can use for it. For all the other symptoms you can find a way to adjust, just as Hawking did in a far worse situation.

This might be one of the reasons for the bad reputation of ME/CFS: there isn’t awareness about cognitive issues, no one talk about them (and in some cases, they are in fact not present at all). But I am pretty sure that the real source of disability in these patients, the lack of productivity, is due to their cognitive problems (when present). Also because in a world like ours, you can work even without using most of your muscles. It is not impossible, I would say it is the rule for a big chunk of the population.

The following one is an interview with Norwegian neurologist Kristian Sommerfelt, in which he points out some analogous considerations. He has done some research on ME/CFS with the group of Fluge and Mella, including the well-known study on pyruvate dehydrogenase. From the subtitles of the video (minute 3:38):

“This [the cognitive problem] is a very typical ME symptom and some of what I believe causes the main limitation. I don’t think the main limitation is that they’re becoming fatigued and exhausted by moving around, walking, running, or having to sit still. If it were just that, I think many ME-patients could have had a much better life. But the problem is that just actively using the mind leads to problems with exactly that, using the mind. It comes to a stop, or slow down, depending on how ill they are.”


¹ It seems that Stephen Hawking had a very unusual presentation of Amyotrophic Lateral Sclerosis (ALS): one half of those with this disease die within 30 months from the first symptom; moreover one out of two ALS patients has a form of cognitive impairment which in some cases can be diagnosed as frontotemporal dementia [R]. So, Stephen Hawking was somehow lucky, in his tragedy, and he doesn’t represent the average ALS patient. I mentioned his case as an example of a person with very severe physical impairment and no apparent cognitive decline, not as an example of the average ALS patient. 

About me

About me

The following video is meant to be a presentation of both the blog and of myself. As I started improving again, some days ago, I decided to record this monologue, so that there could be a video memory of my struggle.

This winter I have spent almost three months in South America, to see if I would have improved during the austral summer, as I usually improve during the Italian summer; and in fact, I did improve. When I came back to Italy (in March) I had a relapse, though. For the last three months, I have been mostly horizontal, without reading or thinking for most of the time.

Now I am climbing the mountain again: I started my rehabilitation reading novels some days ago, then I switched to simple calculations and now I have written my first small code since March. And when I will reach the cognitive level I had about 20 years ago just before I got sick, I will lose everything for months (or years) and I will have to wait without thinking much (despite my best efforts) until I can start all over again…

The myth of Sisyphus has been shaped after me.

 

 

My saviour

One of my short-lived summer improvements (2013). During all these years, as soon as I started feeling better, I opened my books, even before taking a shower and having my hair cut. Happy as a child for most of the time, but also profoundly saddened for the time lost, especially at the beginning of the improvement, when I could realize how much time had passed from the previous positive phase.

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Each time I had to start exactly from where I had left many months or even years before (the longest gap has been 5 years without studying). I had to do a cognitive rehabilitation each time, learn again how to read properly, how to do math, how to discipline my thoughts, how to code. It is a hard process each time. And then, a few weeks after, when I recovered enough to function mentally, I relapsed again.

I am pretty sure that only this complete, obsessive devotion to studying has saved me from very bad cognitive disability.

Before getting sick, coding and math had taught me how to think. Then, when I became ill, each equation I wrote, each drawing and code, all those efforts made to bring my soul back from wherever it was, they kept me alive for all these years.

My first time with LaTeX

Each mathematical formula I have used in my blog till now was an image: I wrote them in a Word document using the equation editor by Microsoft (that I like very much), then I captured the screen and saved it as an image and then I uploaded them in my blog.

In the past, I tried to find a way to write them as text, without succeeding. But in fact, it is possible, using the LaTeX syntax and a few HTML commands.

These are the first “true” mathematical formulae present in this blog. I will use this page to test the LaTeX syntax for mathematical expressions (a handbook is available here).

\int_{A_1}\int_{A_2}\frac{\partial \Psi(x,y)}{\partial x}dxdy

i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>

A symbol within the paragraph: \pi. A inequality within the paragraph: x_i>0. This is an example of an expression with a pedix that has a pedix: f_{X_1}.

The time machine

I am aware that these are just messages floating in the silence, stored somewhere in the planet as binary numbers. I am writing to myself, mainly, from my remote hiding place.

I have travelled through ages, without really being part of them. All alone with my problem. As a patient with a rare disease that doesn’t even have a proper description, I do not belong to humankind.

But humans have paradoxical behaviours, they care more about a man who lived five thousand years ago in the north of Italy, trapped in the ice of our highest mountains, than of clochards that live right now in pain and loneliness in their community. So it might be that generations from now, someone will find these notes, an archaeologist who will try to build my story, from fragments of what I left behind: drawings and calculations. Mathematics is a universal language, after all, and to some extent, even art is universal; not always but often, good art is forever.

If I fail my mission, history will never record my existence. But it might be that at some point in the future someone will find these notes frozen in the ice of a planet long forgotten by humans themselves, as we now have forgotten Africa, the place we all come from.

 

Why I study my own disease

Why I study my own disease

A lot of patients have asked me why I use my little energies to study my disease, instead of just waiting for science to conquer it.

There are many reasons, the first one being that I am desperate because of the cognitive disability that is worse than death. I am not concerned about the physical limitations, at all, even though I have been mostly housebound for the last 20 years. Another reason is that I like computational biology, and I started studying engineering before getting sick with the idea of switching to bioengineering after graduating. So, this is my job.

The other point is that even though I fit the criteria for ME/CFS, I have a rare disease, granted that ME/CFS is not a rare disease: it has a prevalence of 0.4% according to some studies, so it is relatively common. Then why am I a rare patient? Let’s do some calculations: since the median age at onset is 36.6 years with a standard deviation of 12.3 years, the proportion of males is 19% and the proportion of those who are housebound for most of their disease is 25% (R), the probability for a ME/CFS patient of having my same characteristics is (assuming that these are all independent random variables) given by p=0.0075. So, less than 1 out of 100 ME/CFS patients has my type of illness. If we consider also that pain or aching in muscles is present in 59% of patients and it is mostly absent in my case, the above-mentioned probability is even lower: p=0.0031. Which means that only 3 ME/CFS patients out of 1’000 have my illness.

Taken all these data together, the prevalence of my disease in the general population becomes 1/100’000, which means that I have a rare disease, according to the European definition (where a disease is defined rare if it has a prevalence ≤1/2’000) [R]. In the US a disease is defined rare only if it has a prevalence ≤1/200’000, though [R].

And besides that, it is pretty obvious that I am a unique case. I have never found a patient like me, so far. It can also be noted that in Italy, given a population of 60 million inhabitants, those who have my condition are only 600. This might be the reason why I haven’t met them, yet.

This means that I am probably the only one who is studying my disease, on the planet! This is why I’m doing what I’m doing.

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Above, my personal book of immunology, built page by page, paper by paper. I have three other books about this discipline. One of them was a gift from a neurologist that perhaps thought that gift was the only thing she could do to save me. Another one is a very sophisticated text on cutting edge immunology. But this one is the best one because I have selected and read each one of its pages. It is by no means a complete book, it is mainly focused on B cells and B cell autoimmunity, but it has been very useful.

I have built several other books like this, on computational methods in immunology, on metabolism, on neurosciences, on microbiology, and on some diseases: Lyme, ME/CFS, mast cell activation, POTS…

I have learned a great deal, even though outside academia. But I had no choice, I have been too sick and too slow to study at university: I had only a few weeks in which I could study, and then months or years in which I had to wait. This has been my routine. Moreover, given the lack of energy and time, I had to study only what was truly important for my health. Because my goal was to cure myself and save me from a lifetime of cognitive disability.