{"responseHeader":{"status":0,"QTime":7,"params":{"q":"{!q.op=AND}id:\"712933\"","hl":"true","hl.simple.post":"","hl.fragsize":"5000","fq":"!embargo_tdt:[NOW TO *]","hl.fl":"ocr_t","hl.method":"unified","wt":"json","hl.simple.pre":""}},"response":{"numFound":1,"start":0,"docs":[{"conference_title_t":"Seventh Conference of Cardiovascular Training Grant Program Directors","ark_t":"ark:/87278/s6bp3bzb","setname_s":"ir_uspace","restricted_i":0,"department_t":"Biomedical Informatics","format_medium_t":"application/pdf","creator_t":"Warner, Homer R.","date_t":"1960","mass_i":1515011812,"description_t":"Biomedical Informatics","first_page_t":"14","rights_management_t":"Copyright © 1960","relation_is_part_of_t":"Homer R. Warner Collection; Biomedical Informatics Collection","title_t":"The Use of an Analog Computer for Analysis of Physiological Systems","id":712933,"publication_type_t":"Conference Paper","parent_i":0,"type_t":"Text","thumb_s":"/55/e7/55e71664dc375228ddf2867e08d89cad35df774c.jpg","last_page_t":"28","oldid_t":"uspace 10950","metadata_cataloger_t":"AMT","format_t":"application/pdf","subject_mesh_t":"Computers, Analog; Mathematics; Systems Biology; Fourier Analysis; Signal Processing, Computer-Assisted; Monitoring, Physiologic","modified_tdt":"2016-06-22T00:00:00Z","school_or_college_t":"School of Medicine","language_t":"eng","file_s":"/f5/31/f531764c9d3325c39d27334daff1dc450ee2daf4.pdf","format_extent_t":"6,452,631 bytes","created_tdt":"2015-04-14T00:00:00Z","_version_":1664094294934290432,"ocr_t":"- 14 - THE USE OF AN ANALOG COMPUTER FOR ANALYSIS OF PHYSIOLOGICAL SYSTEMS by Homer R. Warner, M.D. Assistant Professor of Physiology University of Utah The subject of this discussion is the analog computer. Since this is basically a mathematical tool, it seems -appropriate to begin with a simple example of a mathematical approach to a physical problem Figure 1. · SYSTEM . DATA tX -,-.. ·X -- ----• t Hypotheses 0 dx -dx -: klt ®-=kt dt dt kt Solution X: X oe X :X o+ 1/2 t2 Linear form of solution In x = k t + constant In x = 2 In t + constant Comparison of data••••• I _::. hypotheses In x L.__ In xt::::_ t ln t • ~ - ..,. - 15 - Let us look at a system which consists of a free falling object in a vacuum. In t his system we · can de ter mine the position of the object at any time. A plot of the dis t ance traveled (x) agai nst time (t) shows that the velocity (which is the s l ope of this plotted line) increaeas with distance and increases with time. Two hypotheses that seem worthy of testing are (1) that the veloc i ty (dx/dt) i s proportional to the dis- ' tance traveled (x), and (2) that the velocity is proportional to the time elapsed (t) since the ob j ect was released. Each of these hypotheses is expressed in the form of a diff erential equation; that is, an equation which expresses the rate of change of one variable with respect to another~ in this case, distance with respect to time. These equations must then be solved. Solving a differential equation such as these two consists of converting it to a form in which one variable is expressed directly as a function of the other. Here, each solution expresses x, the position of the object, as a function of t i me (t). In this form, however, it is difficult to determine the particul ar values for the constants or parameters of the equations that should be used' in order to make a valid comparison between the observed distance and the predicted distance traveled by the object at any time. Fortunately both of these equations can be reduced to a linear form making this comparison easy. A plot of the logarithm of the measured dist ance against time fails to fit a straight line as would be predicted from hypothesis #1 . However, a straight line does fit the data when the logari thm of x is plotted against the logarithm of time as predicted by our second hypothesis. Having thus established equation #2 as describing the data, we may then, by a further manipulation of the original equation; namely, differentiation, arrive at the fundamental conclusion that in this system acceleration is constant. The steps involved in this analysis are common to the analysis of many systems. These steps are reviewed . in Figure 2. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Quantitative observations Observations suggest hypothesis Hypothesis expressed as differential Solut ion of differential equation data equation Comparison of solution with Deduction and prediction f rom equation - 16 - First, quantitative observations are made of the relationship of one or more variables in the system to other variables including time. (2) These observations may suggest an hypothesis or generalization regarding the system being studied. (3) This hypothesis is expressed mathematically, usually in the form of a differential equation. A differential equation again is an equation which includes one or more derivatives; that is, terms which express the rate of change of one variable with respect to another. (4) The differential equations are solved in order to obtain an expression in which the equation variables are in the same form as the experimental data they are to represent. This usually means elimination of the derivative terms by integration. (5) The solution is compared with the experimental data . This involves finding the optimal set of equation parameters.· In the illustration just shown this was done by reducing the solutions to linear forms and then comparing the data to these straight lines. In more complex systems it is necessary to obtain ·repeated solutions, each time varying the parameters until the set of parameters is found that allows best prediction of the data by the equation. (6) And finally, through examination and further manipulation of the equation, general properties of the system may be deduced which might not have been apparent except through this formal expression of the hypothesis. Also the equation may suggest directions in which future studies should be aimed in order to further test the hypothesis . This type of mathematical approach has not been widely used in physiology in the past for several reasons. The biological scientist has been, and to some extent still is, limited in his ability to measure some variables with the accuracy necessary for an adequate analysis. In addition, the mathematical expressions of hypotheses devised to explain physiological systems are often non-linear simul taneous differential equations which may be difficul t or impossible even for the most able mathemati.cian to solve analytically . It is in overcoming this second obstacle that the analog computer can be of valu~b l e assistance to the physiologist. What is an analog computer? I n the example just presented we considered the concept of describing a physical system by an abstract or mathematical expression. Since this is possible, it is conceivable - 17 - that a physical system might be devised to represent any particular mathematical expression. Devising such a system might be of value if (1) the components of the·physical system which represent the parameters of the equation could be easily manipulated and (2) i f the relationship among the variables in the physical system could be directly observed. Of the physical systems. which have been devised for this pur- ' pose electronic analog computer has been the most useful. To provide some insight into how such a computer operates let us look at the diagram shoWll in Figure 3. input= · Addition Integration Oi ff erentiotion O-e5 ooutpul -Ke 5 =- The heart of an analog computer is the operational amplifier , three of which are represented · here as triangles. To be an effective element in a computer such an amplifier must have certain characteristics, among which high gain and low drift are particularly important. The input end of the amplifier is represented by the blunt end of the triangle and the output by the other ~ By selecting the elements in the input and feedback circuits, the amplifier can be made to perform several different types of mathematical operati ons such as addition, integration, and di fferentiation. For instance, if voltages e1 and ez, having the time-course shown, were fed into amplifier number 1, its output e3 would equal t he sum of these voltages. Notice that a change of sign occurs at each amplifier. If e3 i 5 - 18 - n~w fed into amplifier #2, which h&s a capacitor in the feedback circuit , its output e4 would be the,integral of e3 with respect to time. The presence of a capacitor in the input circuit of ~plifier 3 results in its output being the derivative of its input. Thus the process of programing an analog computer consists of first reducing the equation to be solved to a form in which it can be represented on the computer in building-block fashion and then selecting the proper combination of computer components to perform the mathematical operations called for by each term in the equation. The only requirement of the input variable is that it be reduced to the form of a varying D.C. voltage. For instance, the input might be the voltage signal from a transducer. This voltage may be fed into the computer directly from the biological system or recorded on magnetic tape and analyzed at a later time. Even data in the form of a plot on a piece of graph paper ~y be translated to a varying voltage signal using one of many devices, called function generators, developed for this purpose. The solution generated by the computer is also in the form of a voltage and may be observed on an oscill oscope or recorded directly. In this way we may represent an equation or hypothesis by a physical system. The equation parameters may be readily manipulated and solutions may be directly displayed on an oscilloscope or recorded. Before leaving Figure 3, I wish to emphasize that the analog computer merely plays the role of a tool for solving the equation. The basic philosophy of the mathematical approach previously discussed is not altered by the use of the computer. With this background let us now consider some ways in which this tool may facilitate the application of mathematical .technics to the analysis of some physiological systems. Our first consideration must be to define the limits of , the system we wish to analyze. Since in most cases this system will be something less than the whole organism there are several alternatives. The usual approach of isolating the system anatomically from the rest of the organism allows the experimenter to control the input to the system and hold certain factors in its environment constant. However, it imposes the difficult task upon the experimenter of maintaining these variables as close as possible to the · values existing in the intact animal if his analysis is to reveal the particular characteristics of this system which are important in determining its performance in the intact organism. Another approach, which I will call the transfer function or inputoutput approach, may also be applied to the analysis of a biological system. In this approach the system is defined as consisting of all components of the organism which play a role in determining the transition from input to output. The transfer function is the ratio of system output to system input expressed in a form which takes into account not only the amplitude changes - 19 - but the temporal relations as well . As phys i o l ogi sts we are familiar with this approach as a me ~ hod of describing the characteristics of a linear transducer, for instance a pressure gauge .. We know that if sine wave variations in pressure of constant amplitude are applied to a pressure gauge the amplitude of the excursions in signal output voltage will depend upon the frequency of the sine wave. In testing the response of a pressur e gauge to s i nusoidal variations in pressure we realize, of course, that the gauge is not going to be subjecteq to pure sinusoidal variations in pressure when it is used to measure pressure in an artery. However, since any complex wave form may be expressed as the sum of sine waves whose frequencies are integral multiples of a fundamental frequency, our testing of the instrument's response to pure sine waves will reveal the information we need in order to evaluate the instrument ' s response to any input . Also ~aving once determined the instrument's transfer function it is a simple matter to write the differential equation which describes its performance. Now, in testing the response of a pressure transducer it is not necessary to use as an input pure sine waves of constant amplitude and varying frequency . Actual l y any wave form might be used for the input. Then by resolving this wave form and the output waveform from the transducer into their component sine waves (a procedure cal led Fourier analysis) the transfer function or input-output relationship for this transducer may be determined . This approach is not commonly used to obtain the transfer function because it is so t ime consuming. However, with the help of an analog computer it is possible to obtair. the transfer function of such a system in a direct manner . This is illustrated in Figure 4. input~ pressure Known Transducer Volt ofe':'igno I proportional to input pressure Test Pressure Gauge ~ t-----......output voltage Computer Pred icted signal from test gauge test guoge output Dual Beam 0 sci lloscope - 20 - Here is shown a block diagram to illustrate the transfer function approach to the study of a system, in this case a pressure gauge. On the computer a differential equation is programmed which represen.ts our hypothesis to explain the way in which the signal voltage from the test gauge will relate to a variation in the pressure which the gauge sees. A second transducer whose characteristics are known to be adequate to faithfully ' generate a signal proportional to this pressure is used to supply the input to the computer. The parameters of the equation are then adjusted empirically until the solution being generated by the computer matches the output signal from the gauge being tested. If the hypothesis is correct a match between the two waveforms will be obtained. When this has been accomplished the parameters of the equation may be read directly from the computer. In this manner the transfer function of the test gauge can be . obtained. Now we will apply this same approach to a biological system. In Figure 5 is shown a segment of the arterial bed. P1 is the pressure wave-input multichannel tap.e loop recorded~ input signal Biological System Transducer Computer output recorded outp~ signal Predicted output signal ( sol utiow) Dual Beam Oscilloscope - 21 - 0 form ent ering thi s segment and P2 the pr essure at the other end--the output. Using hi gh fide l i t y transducer s these pr essur e s are convert ed to voltages wi thout di s t or tion . The hypothes i s used to explain the change in contour of a pr essure wave traveling down a segment of art:·:!ry is based on the fact that the wall of the artery has inert ia and is distensible and that some of t he pressure i s dissipated as friction , These three properties are treated as paramet ers of a 2nd order differential equation shown here and this equation programmed on the computer. It was found that this equati on will predict P2 from P1 i n many segments of the arterial bed with an accur acy that i s surprising consi dering t he fact that the mode l of the segment of artery i s such a s i mp l e one . Det ermination of the particular transfer function of any given segment of artery permits characteri zation of t hat segment in t erms of a resonant frequency and a damping coefficient. Two features of thi s appr oach dese rv~ emphasis. First, an anal ys i s of a system within the or ganism was performed without having to either add an artificial input to the system or r emove the system from its normal relationship with t he rest of the or gan'ism. And second, this approach is not limited to systems whose input i s periodic. In Figure 6 is illustrated one technic for reproducing a trans ient phenomenon many times making it subject to the type of anal ysis j ust described . The input and output data input pressure (P,)\"'\\ Segment of Arte ry Transducer Tran sducer b d p2 + c P.2. = p d t I Dual Beam Oscilloscope - 22 - from the biological system are converted to electrical form by transducers and recorded on a continuous loop of multichannel magnetic tape. The data may then be reproduced ovet and over again. The input signal is fed to the computer where the computer modifies this function in accordance w~ th the equation being tested and generates an output which can be compared on an oscilloscope with the recorded output signal from the tape. This allows a solution to be obtained with each revolution of the tape loop, permitting emperical adjustment of the equation parameters to finally achieve the best possi ble match between the solution and the observed output, As an example, this particular approach has been successfully applied to the analysis of the time-course of the specific activity of circulating gramulocytes following the injection of a radioactive label. 1 Now let us turn our a t tention to another type of physiological problem which an analog computer may assist in solving. This is the problem of understanding the means by which a biological system regulates its output within certain well-defined limits despite large variations in input to the system. Although physiologists have long been concerned with this problem few quantitative studies have been undertaken. Since World War II the science of control engineering has made rapid strides in the development of technics for analysis an.d design of control systems. Many of these technics might be applied with advantage to the study of control in biological systems. ~ Any control system is a closed loop ; that is, information regarding the output of the system is fed back in some way to the input. An analysis of such a system consists of evaluating the role played by each component of the loop in determining performance of the system as a whole. As an example let us consider the circulation. · Here the analog computer has been used to advantage in analysing the regulation .of the circulation that occurs · in an animal deprived of reflex activity. The intrinsic control readily demonstrated in such an animal must be expl ained 'by the interrelationships among the physical properties of the circulatory system. This might be termed the \"auto-regulation\" of the circulation . An analysis of this autoregulation must be performed before it is possible 'to properly evaluate the role of various reflexes in the regulation of the intact system. Although such an analysis has been performed, only an outline will be presented here in order to. bring out ce~tain features of this problem which may be of gene.ral interest . - 23 - BLOCK DIAGRAM OF CIRCULATION Left Atrium ~----.~--iii>DOnd ----r Pulmonary artery I~ systole diastole P ulmonary veins '~ diastole systole Left ventricle ~ · Arterial bed I Rightvf~n-t-r-ic-le----~R~·~ig_h_t_a_t_n_·u~m [ ton dollllii:<~.--.! Systemic veins Figure 7 shows a block diagram of the circulation which consists of 2 pumps, 2 transmission lines, and 2 reservoirs in a closed loop. Much information is available concerning the characteristics of each component part of this system. From this information a set of 16 equations were derived to represent the interrelationship of volume, pressure, flow and time in each part of the system. Simultaneous solution of these equations allows prediction of the behavior of the whole system. If the system is disturbed from its state of equilibrium (for instance by displacing blood from the pulmonary circulation into the systemic circulation as occurs during a Valsalva maneuver) the time course of each variable may be observed as the system returns to equilibrium and this response compared to the response observed in the dog deprived of reflex· activity. The parameters of the equation are adjusted until the solutions obtained accurately dupli cate the biological observations. Manipulation of the equation parameters allows quantative assessment of the role of each component part in determining the performance of the system as a whole. In this way a framework is established upon which an analysis of the role of various reflexes in cardiovascular control may be undertaken. - 24 - In some situations it may be desirable to represent only one component of a closed loop or self-regulating system on the computer. An example of this approach is shown in Figure 8 . {input volt= I< 1 P) ( Pressure Transducer Carotid Sinus ~Nerve ~----~f ~'-ti _m_u_al -to-r f = l\\3eo Computer ~ (output voltage= K2f) Carotid Sinus Equation f =a~~ +b(p-po) In this example the computer is subs t ituted for part of a control system, The equation shown here is derived from available data concerning the relationship between input and output of the carotid sinus and is pro-gramed on the computer. The equation states that the frequency of action potentials (f) traveling up the carotid sinus nerve is proportional to the rate of change of pressure in the carotid sinus and to the pressure itself when it is above a certain threshold pressure (P0 ) . The input to the computer is a voltage signal (ei) from a strain gauge sensing arterial pressure in a carotid artery : The output from the computer (e0 ) drives a stimulator whose frequency is proportional to this voltage. The output of the stimulator is connected to electrodes on the afferent end of a previously severed carotid sinus nerve. Thus a rise in arterial pressure results in I .. - 25 - an increased rate of car otid s i nus nerve stimulation just as would occur with the carotid sinus itsel f. With the computer, however, it is now possible to alter at will any one of the equation parameters (a,b, or P0 ) and study the effect of thi s change on the behavior of the circulatory system. It was found that i ncreasing t he steady state gain (b) re- , sulted in oscillations in arter i al pressure similar to the so-calle~ Traube-Hering waves. These oscillations are attributed to a phase change resulting from lags in the response of arteri al smooth muscle to changing sympathetic efferent activity. Thus , t his example illustrates that substitution of the computer for a component part of a control system will allow quantitative evaluation of t he role of this component and may bring out certain characteristics of the r e s t of the system that might be difficult to detect otherwise . Finally let us consider t he use of a computer to gatn insight into the kinetics of physical and chemical processes occuring at the cellular •level from data obtained from a relativel y intact gross preparation. As an example we consider the regulation of heart rate by efferent sympathetic action potentials (Figure 9). CEL LS OF S.A. NODE Sympathetic Nerve f A0 CD [Ao] = k 1 f n @ d [Ai] = k (A -A ·)- d[AB] dt 2 O I dt r-2'1 d [AB] _ . . _ \\2J dt - k3 [A 1)[BJ k4 [.A BJ @) [B] + [AB] = [Af!j max . @ 11HR = k5 [AB] Recorded H.R. 4 f I sec. 0 Predicted H.R. · · · ., \"· - .. ·r · · ·,::.~·._ .. ·· . ..:;.· ~~: ... /~;~~· , . ~r,\"'lfj\"'\"\" , •, ; ~ . .... ' . .. ....... ... ' I . ~ ... . ~, .. ,· :·lflfli/· ·:·. I.,.~ .. ... _ . • .. ·I I • • ' ... ' • ! ~ -i t- 8 sec. -240/min . -140/min . +Recorded H.R . - 26 - • I The heart rate is determined by the frequency of action potentials in one or more cells in the region of the S.A. node in the right atrium. Modification of this rate ~ay be effected by variation in efferent activity of the sympathetic nerves which end in this region. Our problem is to find an hypothesis or equation which will predict heart rate (HR) from the fre; quency (f) of action potentials on cardiac sympathetic nerves in the absence of vagal activity. Both vagus and sympathetic nerves to the heart were severed. (f) was varied by varying the rate of stimulation of the cardiac sympathetic nerves. The heart rate with each cycle was calculated by the computer from the interval between R waves of the electrocardiogram. Voltages proportional to input (f) and the output (heart rate) were recorded on magnetic tape. In the top recording is shown the observed heart rate response to a step increase and decrease in frequency of stimulation of sympathetic efferents to the heart . The following characteristics of the response of this system were observed . No change in heart rate occurs for 1.5 seconds following a step increase in f . Heart r ate then rises to a new level. Over the range of stimulation frequencies in which these small sympathetic fibers will respond increasing f shortens the time required to reach this new level. The increment in heart rate achieved when the heart rate reaches its new level is not proportional t .o f but .asymptomatically approaches some maximum value. When f is suddenly decreased, heart rate falls much more slowly than it rose. From such observation's the Kinetic model and system of equations shown here was derived. Equation 1 states that the extracellular concentration of noradrenaline (Ao) is proportional to the frequency of stimulation (f) times the number of sympathetic fibers responding (n). According to equation 2, the rate at which the intracellular concentration of noradrenalin (Al) changes with time depends upon the rate of diffusion of water across the cell membrane (k2), the difference between the inside and outside concentration of noradrenalin and the rate at which noradrenalin is combining with some intracellular substance to form compound AB. Equation 3 expresses the rate at which compound AB is being formed. This is a second order reAction in which the rate depends upon the concentration of noradrenalin and another substance (B) which reacts with it. We assume that the number of molecules of B present is a constant as shown in equation 4. This limits the amount of AB that can be formed and thus the maximum change in heart rate that can be achieved, since change in heart rate is proportional to (AB) concentration as shown in the last equation. That this hypothesis will predict the heart rate response to a variety of patterns of variation in frequency of nerve stimulation has been demonstrated using the approach shown here. The proper values for the equation coefficients are found by adjustment of the corresponding elements in the computer to values which will make the time course of heart rate predicted - 27 - by the equations correspond to the r ecorded heart rate response . The predicted heart rate in response to this step increase in f i s shown i n the middle frame, and in tne bottom frame the recorded and predicted response are superimposed . To es t ablish the constants of these equations one must match the heart rate r esponse to two step inputs of dif ferent magnitude. Once this is accomplished for a given dog, these equations will predict the time-course of heart rate resulting from any subsequent input pattern of variation in f. I would like t o point out that in deriving an equation for a system such as this, the system is not treated strictly as an unknown entity or black box . All known facts from any source are used. This is as it should be since the equation or hypothesis must account for these facts as well as those being observed in the present experiment . The important point is that the kinetics of this physical-chemical process observed in a r e latively intact gross preparati on may be evaluated in a quantitative manner with the help of the computer . Finally, two other points (in regard to this approach) should be emphasized. First, when an equation is found which will describe a system, its value must be judged on the basis of .its ability to describe the system under all circumstances. The equation should have a minimum number of parameters and each of the parameters should be s ensitive to changes in a particular system characteristic. If these two criteria be satisfied, the que~ uniqueness of the equation ~ ... \"Tile second pOID1:(ieserving em~s the -~ct that valuable information may be obtained each time an equation fails to predict the behavior of a system. Such a failure means that the concepts regarding this system's performance are inadequate to account for the observed facts since the equation being tested was derived from these concepts. Thus a modification of the prevailing concepts is necessary and new concepts must be sought. It is my opinion that this \"negative information\" may be t he most valuable type of answer that the computer can give us. To summarize, a simple physical problem was first presented as a means for outlining the steps involved in the applicat i on of mathematics to the analysis of experimental data . The concept of a transfer function or input-output relationship was introduced , first through an example of its use in the evaluation of a pressure transducer and then by applying this approach to the study of pressure wave transmission by a segment of the arterial bed. The role of the computer in the analysis of closed-loop or self-regulating sys t ems was illustrated by 2 approaches- -one in which the performance of the whole system was expressed on the computer by the simultaneous solution of a set of differential equations and the other in which the computer was substituted in a biological system for one element of this system. Finally, an example was presented to i llustrate the use of the computer in deriving information from a relatively intact gross preparation regarding the kinetics of physi cal and chemical processes - 28 - '·at the cellular level. The analog computer is a useful tool for the physiologist who would apply mathematical technics to the analysis of the systems he studies. concepts to bio=medical s c ience is dual; we as well as the hardware for applying t hem."}]},"highlighting":{"712933":{"ocr_t":[]}}}