retrospectroscope

medical alchemists who called their prac- The Mathematization tice iatrochemistry, meaning the chem- istry of healing (should we consider it of Biology and Medicine an ancestor of biochemistry?) and is the source of the modern word ped-iatrics. Who, When, How? Thus, iatromechanics would be the prede- cessor of our current biomechanics. Some- By Max E. Valentinuzzi and Alberto J. Kohen how, Borelli followed the mechanistic philosophy inaugurated Nessuna umana investigazione si to mathematize biology by René Descartes (1596– puo dimandare vera scienza s’essa and medicine from da The objective herein is 1650), that is, believing non passa per le matematiche Vinci’s time to the pres- to review and collect that living things behave dimostrazione. ent, where mathematical the most important like machines or artifacts. —Leonardo da Vinci (1452–1519) biology (often including Philosophers of science by sheer need) efforts made to call such a view a reduc- ne objective of bioengineering is the has become an estab- mathematize biology tionist approach, that quantification of the biological and lished discipline with and medicine from is, attempting to reduce Omedical sciences, with the goal of departments, congresses, da Vinci’s time to complex biological sys- improving their exactness and precise- and journals devoted to the present. tems to simpler ones (such ness, always in an attempt to remove as its further development. as mechanical, hydraulic, many indeterminations and uncertain- No doubt, it supplies an or electrical systems). ties as possible, especially when seeking essential background to bioengineering. Perhaps other examples could be predictions. The extract at the begin- It must be mentioned that the initial found, like the simple numerical thinking ning of the column, which is attributed chronology and a few concepts were of William Harvey, who on the basis of ani- to Leonardo da Vinci, seems to reflect a taken from the introduction of Máximo mal experimentation, asserted that blood very early Renaissance concept in such Valentinuzzi’s doctoral dissertation (Max moves in a closed circuit, accepting this fact a direction; this concept is indeed more E. Valentinuzzi’s father [2]). A short his- as a biomathematical achievement. Luigi general because it encompasses all the sci- torical summary can also be found in [3]. Galvani (1737–1798), an experimental ences [1]. The famous Vitruvian Man, so biophysicist and the discoverer of animal many times reproduced on book covers Early Attempts electricity, might be included as a fore- and posters, appears perhaps as an antici- Giovanni Alfonso Borelli (1608–1679) father of biomathematics, for many were patory geometrical epitome of that mod- was an Italian physicist and mathemati- the mathematical trials and developments ern recent aim (Figure 1). However, 200 cian who made significant contributions that much later explored electrophysiol- years before da Vinci was Ramon Llull to medicine [4]. In the city of Pisa, Italy, ogy. We must philosophically mention (ca. 1232–1315), a Majorcan philoso- he met Marcello Malpighi (1628–1694), a the contributions made by Carl Ludwig’s pher, member of the Third Order of Saint physician who encouraged him to get into school during the 19th century as well as Francis, and author of important works medical subjects (by the way, Malpighi is that of several of his followers and disci- of Catalan literature. Lull is considered well known for the discovery and descrip- ples [5]. They certainly gave a significant by some to be a pioneer of computation tion of the pulmonary capillaries, thus impulse to the quantification of physiology theory because of his contributions to log- completing what William Harvey (1578– and medicine; however, the true ics. Should he also be considered a very 1657) had anticipated regarding the blood mathematical model of a biological system early predecessor of biomathematics? We pathway as a closed circuit). As a result, had to await more maturation. leave this question dangling for the his- and after studies and experimentation, torian and/or the philosopher of history. Borelli came up with an opera magna, The 20th Century In Spain, there is a journal carrying his De Motu Animalium (On Animal Movement), name, Llull (see [12]). which was published two years after his Alfred James Lotka, Vito Volterra, The objective herein is to review and death. The book deals with mechanics Pierre Francois Verhulst, and collect the most important efforts made and mathematics as applied to medicine. Vladimir Alexandrovitch Kostitzin He used the term iatromechanics (iatros is Perhaps the first significant contribution,

Digital Object Identifier 10.1109/MPUL.2012.2228591 the ancient Greek word for physician). systematically starting the mathematiza- Date of publication: 13 February 2013 This root was later used by the medieval tion of biology, belongs to Alfred James

50 ieee pulse ▼ January/February 2013 Lotka (1880–1949), who was born in Lviv (which was at that time situated in Aus- tria and is now located in Ukraine) and died in the , where he moved in 1902. He is the author of a number of theoretical articles on chemical oscilla- tions during the early decades of the 20th century and of the first book on theoreti- cal biology, which appeared in 1925 [6], [7]. That book, full of deep philosophi- cal–epistemological citations and digres- sions and not an easy read, indeed, goes into difficult areas such as consciousness and the concept of life itself. Lotka is best known for the predator–prey model, now called the Lotka-Volterra model, which he developed independently from Vito Volterra (1860–1940). This model is still the basis of many others, as, for example, in such odd fields (at least for us bioen- gineers) as the relatively recent re-engi- neering of corporations [8]. Vito Volterra, 20 years older than Lotka, was an Italian mathematician and physicist. After World War I, he turned his attention to the application of math- ematics from biology, principally reiterat- ing and developing the previous work of Pierre Francois Verhulst [9]. The Lotka– Volterra equations, or predator–prey equations [10], are a pair of first-order, nonlinear differential equations describ- ing the dynamics of certain biological systems. Two species interact: one is a predator and the other is its prey; they can refer to bacteria, insects, or any other Figure 1 Leonardo da Vinci’s Vitruvian Man is based on the specifications given kind of animal that responds to such by Vitruvius in De Architectura (first century BC). da Vinci tried to find the perfectly pro- competitive interaction. Something of portioned body. It was created circa 1487, with pen and ink on paper, depicting a male the type would also be the acetylcholine- figure in two superimposed positions with his arms and legs apart and simultaneously inscribed in a circle and square. Stored in the Gallerie dell’Accademia, Venice, Italy. acetylcholinesterase kinetics at the myo- neural junctions. They evolve in time according to the hypothesis that states that the rate of growth of the prey is pro- portional to the number of prey units where Verhulst in 1838 came up with the follow- minus a nonlinear term that includes ▼▼ x is the number of prey (say, mosqui- ing equation: the number of prey units and the num- toes in a swamp) ber of predator pieces; besides, the rate of ▼▼ y is the number of predators (for dN =-rN 1 N , (3) dt K growth of predators is accepted as being example, frogs in the same swamp) ` j negatively proportional to the number ▼▼ dy/dt and dx/dt represent the growth where N(t) represents the number of indi- of predator units plus a nonlinear term of the two populations against time viduals at time t, r is the intrinsic growth including both prey and predator units, ▼▼ t stands for time rate, and K is the carrying capacity or that is, ▼▼ ab,,c, and d are parameters taking the maximum number of individuals an into account the interaction of the environment can support. He called its dx two species. solution the logistic function, the name =-xy()ab (1) dt And who was Pierre François Verhulst? that is still kept, that is, dy Born in Belgium in 1804, he died young K =-yx()cd- , (2) Nt()= , (4) dt in 1849. An accomplished mathematician, 1 +CKe-rt

January/February 2013 ▼ ieee pulse 51 organic compounds, carbon, oxygen, and nitrogen were considered. One chapter is devoted to the logistic equation, which had already been introduced by Verhulst long before (see above), with other chapters dealing with growth at large, populations, milieu (environment, cul- ture broth), relationships between species, symbiosis and parasitism, embrionary growth, the shape of living organisms, and evolution, that is, all very funda- mental and purely biological subjects (Figures 2 and 3). But there is much more to be said about Volterra and Kostitzin. Giorgio Israel and Ana Lillán Gasca, in a com- prehensive and superb paper done at the Universitá di Roma—La Sapienza, Italy [12], explain well the relationship between Volterra and Kostitzin in an overview of the biomathematical con- cepts of the 1930s, which was in many respects a worldly dramatic decade, along with the intervention of other distin- guished scientists of those days. From the somewhat translated, paraphrased, and much summarized information in the report by Israel and Gasca, let us proceed by saying that Volterra offered during the winter of 1928–1929 a course on bio- mathematics at the Institut Henri Poin- caré, Paris. Those lessons were published in 1931 under the French title Leçons sur la Théorie Mathématique de la Lutte pour la Vie (in English, Lessons on the Mathemati- cal Theory of the Struggle for Life). There are Figure 2 The front page of Kostitzin’s book Biologie Mathématique. This image was taken from a copy belonging to the author’s personal library, bought by the author’s international bookstores, which can be father, Máximo Valentinuzzi, who was an active biomathematician and biophysicist found on the Internet, that offer copies of who spent several years with Nicolas Rashevsky in Chicago in the late 1950s and early either the original publication or a more 1960s. (Image courtesy of Max E. Valentinuzzi.) recent reprinted issue. Volterra studied the evolution of pop- where C =-10//NK1 is determined training, he produced a book on math- ulations of different species that share ^h by the initial condition N(0). More infor- ematical biology, originally written in a medium and a biological association, mation about the latter equation can French, with a preface by Vito Volterra such as competition for food or a prey– be found either on the Internet or in [11]. In the preface, Volterra says that, predator or parasite–host relationship. appropriate textbooks. after a number of papers on this subject up The reductionist approach triggered much Vladimir Alexandro- to its publication in 1937, discussion and objection. The long corre- vitch Kostitzin must be there was no other such spondence between Volterra and Kostitzin mentioned as a prominent Lotka is best known didactic and complete contains moving testimony. They met scientist, though perhaps for the predator–prey piece collecting essential when, for unknown reasons (perhaps not well recognized. He knowledge where general political), Kostitzin and his wife, Julie (a was born in 1883 in Rus- model, now called the mathematical notions, parasitologist who got a professorship at sia, although his death Lotka-Volterra model, frequency curves, corre­ La Sorbonne), moved to . Kostitzin does not seem to have which he developed lations, differential equa- considered Volterra to be his teacher, been registered (perhaps independently from tions, and many basic and together they initiated a rebirth it occurred in Paris in Vito Volterra. biological phenomena of Darwinism, clashing against many 1963). A mathematician by such as circulation of opposing positions. Helped by Volterra,

52 ieee pulse ▼ January/February 2013 Kostitzin was appointed Chargé de Réchérches at the Conseil National des Réchérches Scientifiques (founded in 1939). During the Second World War (1939–1945), Kostitzin was taken by the Germans to a concentration camp, where he remained from 22 June 1941 until 3 March 1943. Recall that France was occupied by the Nazi Regime. Things improved a little after his liberation, but the situation was still difficult. After the war was over, he almost fully abandoned biomathematics. His wife’s demise in 1950 also left a deep mark. Kostitzin’s last letter was to Virginia Almagiá, Volterra’s widow, in December 1962. The Kostitzin– Volterra correspondence is kept in the Accademia dei Lincei (Rome). The letters are written in French, and biomathemat- ics is the central subject, although they also touch on other general questions of the times. All in all, they reflect the tremendous scientific and human per- sonalities of both. For example, one of the letters states Mathematics have got into the nat- ural sciences via statistics, but this phase must yield to the analytic phase, as it happened in the ratio- nal disciplines. Thus, the statistical method clears the field, it estab- lishes some empirical laws and so it makes easier the entrance of the analytical variables. In Kostitzin’s case, there was a coex- istence of the mathematician and the Figure 3 The third page of Kostitzin’s book. Máximo Valentinuzzi’s signature naturalist, always searching for experi- is shown indicating the date on which it was obtained: 13 June 1940. It clearly mental validation. Often, he and Volterra depicts the beginning of the preface written by Vito Volterra. Volterra’s and showed understanding of the rejection Kostitzin’s texts were the first devoted to mathematical biology. (Image courtesy of Max E. Valentinuzzi.) faced by some biologists. Kostitzin kept his optimism in spite of the difficulties, as he pointed out, “in a time when the world changes but does not improve” populations, particularly those of phenotypes in his models to arrive [13]. Kostitzin, unfortunately, never natural selection, mutation, genetic at similar conclusions to the Fisher– received full recognition of his outstand- drift, and gene flow. From these as Haldane–Wright position. ing and remarkable contributions to well as other contributions, there is mathematical biology [13]. These authors a widespread miscon- Balthasar van der Pol of [13] bluntly recall ception that evolution Balthasar van der Pol the historiographic tradition re­­ is a change in gene In Kostitzin’s case, (1889–1959) deserves garding evolutionary synthesis, frequencies. Within there was a coexistence a special section, as he, which emphasized the names of this context, articles with his collaborator van Ronald A. Fisher, John S. Burdon postulating changes of the mathematician der Mark, specifically Haldane, and Sewall Wright as in the frequencies of and the naturalist, dealt with the heart as the the theoreticians who laid the phenotypes (not genes) always searching objective of a model based quantitative foundations of the were not well received. for experimental on oscillations by relax- mechanisms of change in gene Kostitzin’s standing validation. ation [14], [15]. He stud- and genotypic frequencies in favored essentially the ied in England under two

January/February 2013 ▼ ieee pulse 53 grand physicists, John Ambrose Fleming then suddenly jumping to a negative level. Nicolas Rashevsky (1899–1972) (1849–1945), remembered for the diode, This cycle is continued indefinitely. The and his Chicago School and Joseph John Thomson (1856–1940), relaxation period is given by Nicholas Rashevsky was the man who sys- remembered for the electron, and spent tematized modern mathematical research 2 all his productive years with Philips Labo- Trelax = ~a/. (7) in the biological and medical sciences. ratories until his retirement. Van der Pol Born in Chernikov, Russia, Rashevsky clearly stood out among the biologists of Hence, the time period, which is studied physics in Kiev, Ukraine, where he his time. determined by the discharge of a capaci- obtained his degree in 1919. After teach- Van der Pol’s relaxation paper referred tor, is called the relaxation time. It was ing in Kiev, Istanbul, , and Prague, to in the above paragraph starts by pointed out that these types of oscilla- Czechoslovakia, he moved to the United refreshing the usual sinusoidal oscilla- tions are found in nature. Van der Pol States in 1924, where he spent a period tions stemmed at a linear second-order actually constructed a four-element in the Research Labs of Westinghouse, differential equation (well known to simple electrical circuit (capacitor, neon in , Pennsylvania. Some- any electrical or mechanical engineering tube, battery, and resistor) how, his interests shifted student), wherein the squared damping and demonstrated several toward biology. He got factor a must be much smaller than the features related to cardiac Nicholas Rashevsky his fundamental training squared natural frequency, ~, or activity. Students were was the man who under Davenport Hooker given a laboratory exer- systematized modern (1887–1965), an anato- a~2211 . (5) cise which required them mathematical mist, and Charles Claude to check how the differ- research in the Guthrie (1880–1963), a The relaxation idea is based on revers- ent elements influenced physiologist and vascular ing the above-mentioned inequality, the output signal. All this, biological and medical surgeon, both active at that is, no doubt, represented one sciences. the medical school at the of the first fully bioengi- University of Pittsburgh a~2222 . (6) neering accomplishments, as it gathered [16]. In 1934, Rashevsky got a position a rather solid theoretical background at the , where he In the latter condition, the system along with a hard model aimed at a spe- later created the outstanding and far- tends initially to jump off from zero to a cific physiological system. Thus, it went reaching Committee on Mathematical positive value, decreasing gradually, and beyond biomathematics. Biology. There are innumerable contri- butions from renowned authors serving as members of the Committee. Several students obtained their doctorate degrees from the university, which disseminated the discipline on national and interna- tional levels. His culture was ample and deep, projected to music, literature, and philosophy. He spoke several languages fluently. Emily, his wife, also a mathema- tician, was his companion throughout his life. Rashevsky—tall, bearded, and patriarchal—always showed a strong and contagious dynamism, displaying good humor, lively comments, and unrelent- ing serious criticism. Often, he dearly and protectively referred to “his mathemati- cal biology family.” He was the founder of the Bulletin of Mathematical Biology, which is still an active and prestigious publication [17], [18]. The 1961 Cullowhee Conference in North Carolina had two contributions outlining the historical development of mathematical biology since its incep- tion: one was by Rashevsky [19] and Figure 4 From left: Nicolas Rashevsky, his wife Emily, and Máximo Valentinuzzi, Sr., another by Max Valentinuzzi, Sr. [20] taken during the Conference on Biomathematics in Cullowhee, North Carolina, on 14–18 August 1961. The photo was taken by Max E. Valentinuzzi, who also participated (Figure 4). Rashevsky’s contribution in that conference. deals with general and specific subjects,

54 ieee pulse ▼ January/February 2013 highlighting the concept that theoretical biology should be the counterpart of the- oretical physics. Valentinuzzi’s contribu- tion presents a definition of mathematical biology as a branch of natural philoso- phy that embeds biological thought into mathematical scheme. It contains a review similar to that by Rashevsky in some aspects, but which includes other less-known researchers such as two Argentinian scientists, Florentino Ameghino (1854–1911) and Ángel Gallardo (1867–1934), and the Italian physicist Luigi Fantappiè (1901–1956) [21]. In 1942, Fantappiè presented to the Accademia d’Italia a theory attempt- ing to unify the physical and the bio- logical world. In 1938, Rashevsky produced the first edition of a significant book on mathematical biology, which compre- hensively collected well-thought knowl- edge and set the framework of the emerging discipline (Figure 5) [22]. One of the many subjects found there refers to excitable tissues, using the so-called two-factor theory, also called the activa- tion–inhibition theory. It is based on two differential equations, or

df =-KI k()ff- 0 (8) dt dk =-MI m()kk- 0 , (9) dt

where K, M, k, and m are constants. In other words, whenever a current flows Figure 5 The front page of the third edition of Rashevsky’s book, which was published through an excitable tissue (nerve or by Dover in 1960. muscle), ions are transported, say of the activating type f and of the inhibiting kind k. It is assumed that the rate of system, pharmacological them or even to offer an change of the activating or excitatory problems, the endo- The currently available exhaustive list. The dis- type and also of the inhibiting kind crine system, the central computational power cipline stands well on its is proportional to the current flow I. nervous system, and some is a tool to solve own feet, offering a solid Besides, any excess of f or k over the abstract speculations and and ever-increasing foun- any mathematical threshold values f0 or k0, respectively, their potential value, as dation to bioengineer- or any deficiency below those levels tends well as a final appendix problem, irrespective ing, biology at large, and to decrease at a rate proportional to the offering mathematical of the complexity, medicine. The currently excess or deficiency. Relationships with derivations. It can still thus, slowly and available computational other similar theories are well covered be used as an orientation steadily improving the power is a tool to solve in addition to a good number of predic- text for an introductory predictive capabilities. any mathematical prob- tions and explanations. course in mathematical lem, irrespective of the Another excellent book, Some Medi- biology [23]. complexity, thus, slowly cal Aspects of Mathematical Biology, must be and steadily improv- mentioned. It is a simpler but very didactic Discussion and Conclusions ing the predictive capabilities. However, book published in 1964, divided into six Many books have been published on these solutions are still quite distant from parts: retention of particulate material in mathematical biology since the 1960s. the extraordinary situations encountered respiratory passages, the cardiovascular It is not the intention here to review in the physical sciences. The scientist

January/February 2013 ▼ ieee pulse 55 behaves as some kind [3] M. E. Valentinuzzi, [15] B. van der Pol and J. van der Mark, of conqueror, endlessly The scientist behaves ”Objetivos de la bioingeni- ”The heartbeat considered as a relax- searching for newer and as some kind of ería,” (in Spanish), in Intro- ation oscillation, and an electrical model better roads to get to the conqueror, endlessly ducción a la Bioingeniería, of the heart,” Archives des Nederdlands objective. As I. Newton Series Mundo Electrónico. Bar- de Physiologie de l’Homme et des Animaux, Kugelman [24], foreword searching for newer celona, Spain: Boixareu Edi- vol. 14, pp. 418–443, 1929. author of Rashevsky’s and better roads to tores, 1988, ch. 1, pp. 3–11. [16] A. Carrell and C. C. Guthrie, ”The trans- book mentioned in [23], get to the objective. [4] [Online]. Available: plantation of veins and organs,” Amer. stated in 1964, “Mathe- http://es.wikipedia.org/wiki/ Med., vol. 10, pp. 1101–1102, 1905. matics is gradually becom- Giovanni_Alfonso_Borelli [17] Bull. Math. Biol., vol. 27, no. suppl. 1, ing the language of medicine.” Good [5] M. E. Valentinuzzi, K. Beneke, and G. E. pp 3–4, 1965; doi: 10.1007/BF02477255. prediction, indeed, because a model as González, ”Ludwig: The bioengineer,” [18] M. Valentinuzzi Sr. and M. E. Valentinuzzi, old as Kotitzin’s is today being used with IEEE Pulse, vol. 3, no. 4, pp. 68–78, 2012. ”Nicolás Rashevsky,” La Semana Médica, success, and we find reports stating [25] [6] A. J. Lotka. (1925). Elements of Physical Biol- vol. 140, no. 25, pp. 737–738, 1972. Kostitzin’s demogenetic model ogy. Baltimore: Williams & Wilkins. [On- [19] N. Rashevsky, ”A bird’s eye view of the describes local interactions between line]. Available: http://ia600307.us.archive. development of mathematical biology,” in three competing pest genotypes org/35/items/elementsofphysic017171mbp/ Proc. Cullowhee Conf. Training in Biomathe- with alleles conferring resistance or elementsofphysic017171mbp.pdf matics, Cullowhee, NC, 1962, pp. 8–19. susceptibility to transgenic plants, [7] [Online]. Available: http://users.telenet. [20] M. Valentinuzzi Sr., ”Notes on the his- the spatial spread of insects being be/ronald.rousseau/html/lotka.html tory of biomathematics,” in Proc. modeled by diffusion. This new [8] T. Modis, ”Genetic re-engineering of cor- Cullowhee Conf. Training in Biomathematics, approach makes it possible to com- porations,” Technol. Forecast. Soc. Change, Cullowhee, NC, 1962, pp. 20–38. bine a spatial demographic model vol. 56, pp. 107–118, 1997. [21] [Online]. Available: http://www-history. of population dynamics with clas- [9] V. Volterra, Leçons sur la théorie mathé- mcs.st-and.ac.uk/Biographies/Fantappie. sical genetic theory. matique de la lutte pour la vie (in French). html The latter paper calls for a slight Paris: Gauthier-Villars, 1931 (Reissued in [22] N. Rashevsky, Mathematical Biophysics: change in Kugelman’s statement by add- 1990 by J. Gabay, Ed.). Physico-Mathematical Foundations of Biol- ing “… the language of medicine and [10] [Online]. Available: http://en.wikipedia. ogy, 3rd ed., vols. 1–2. New York: Dover, biology.” Unfortunately, Kostitzin’s work org/wiki/Lotka%E2%80%93Volterra_ 1960. is poorly known today, as the authors of equation. [23] N. Rashevsky, Some Medical Aspects of [25] have complained. [11] V. A. Kostitzin, Biologie Mathématique Mathematical Biology. Springfield, IL: (preface by V. Volterra). Paris: Libraire Charles C. Thomas, 1964. Max E. Valentinuzzi (maxvalentinuzzi@ Armand Colin, 1937. [24] J. I. Wasserman. (2012). From Shtetl to ieee.org) and Alberto J. Kohen (ajkohen@ [12] G. Israel and A. Millán Gasca, ”La corre- Park Avenue: Isaac Newton Kugelmass, yahoo.com) are with the Instituto de spondencia entre Vladimir A. Kostitzin M.D. (1896–1979). [Online]. Available: Ingeniería Biomédica (IIBM), Universidad de y Vito Volterra (1933–1962) y los inicios http://www.janetwasserman.com/ Buenos Aires (UBA). de la biomatemática,” (in Spanish), Llull, from-shtetl-to-park-avenue-i-newton- vol. 16, pp. 159–224, 1993. kugelmass-md-1896–1979.html References [13] A. Mellender de Araújo, ”Vladimir [25] Y. Tyutyunov, E. Zhadanovskaya, D. [1] L. Carlo Pedretti. (1999). Le Macchine. Fi- A. Kostitzin, teórico, ignorado pelos Bourguet and R. Arditi, ”Landscape renze, Italy: Giunti Gruppo Editoriale [On- arquitetos da síntese evolutiva,” (in Por- refuges delay resistance of the Euro- line]. Available: http://books.google.com. tuguese), Filosofia e História da Biologia pean corn borer to Bt-maize: A demo- [2] M. Valentinuzzi Sr., ”Contribución al (Brazil), vol. 2, pp. 5–22, 2007. genetic dynamic model,” Theoretical estudio físico de la contracción uterina,” [14] B. van der Pol, ”On relaxation oscilla- Population Biology ­(Elsevier), vol. 74, no. (in Spanish), Ph.D. dissertation, Medical tions,” Jahrbuch der drahtlosen Telegraphie, 1, pp. 138–146, 2008. School, Univ. Buenos Aires, 1950. vol. 28, p. 178, 1926.

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