@MAKE(Report)          @DEVICE(lpt)
@STYLE(PAPERWIDTH=33CM,LINEWIDTH=31CM,PAPERLENGTH=36CM,INDENTATION=10)
          @BEGIN(TITLEPAGE)
Advanced Technology Research Center | Silab Division | Samani International Enterprises 
FERMENTATION GROWTH STAGE DETERMINATION: Fourier Series Fit via Marquardt-Levenberg Algorithm
VASOS-PETER JOHN PANAGIOTOPOULOS II [President, Samani International Enterprises]
                    @VALUE(DATE)
                    @begin(researchcredit)
                              @center(@b(ABSTRACT))
	Bacteria are extensively utilised in some newer chemical processes.
Often, one would wish to approximate the stage of their growth,  especially
so that one can monitor and control by computer.  A
numerical algorithm is proposed  for such determination.
                    @end(researchcredit)          
          @END(TITLEPAGE)
@PAGEHEADING( left "Fermentation Growth Stage Determination",
	  CENTER "@value(page)",
          RIGHT "Vasos-Peter John Panagiotopoulos II")
@PAGEFOOTING( left "@value(sectiontitle)",CENTER "@value(date)",
	  RIGHT "Samani Silab ATRC" )
@Chap(Determination of the Characteristic Curve )          
          Fitting the expected growth curve @foot{T.J.Parry & R.K.Fausey, 
@i,p.19,
UK:Hutchison Educational,1973} to an even Fourier cosine series yields
the following coeffiecients: 
@equation(a@-<0>=9.1325;a@-<2>=-.1756;a@-<4>=-.1496)
@foot[Hewlett-Packard HP41C @i, Program Library ROM, Fourier
Series Program,Corvallis,OR:HP,1979.] 
          These are taken as initial guesses to fit the arbitrarily scaled
data and to determine the arbitrary period, L. This is performed with
Marquardt-Levenberg optimisation methods applied to 
curvefittng [coefficients are the variables over which the optimisation of 
error form data occurs] @foot{ cf: "Iterative Solution of Nonlinear
Boundary Value Problems in Ordinary Differential Equations", Kenneth J.
Anselmo and Jordan L. Spence, Columbia University Department of Chemical
Engineering and Applied Chemistry. Received from Prof. Spencer in
March, 1982.} on a
modeling package. @Foot[Knott, G. D., &
Reece, D. K., MLAB: A Civilized Curve-Fitting System,@i,@b<1>,p.497-526,Brunel,UK(1972); Knott, G.D.,&
Reece, D. K.,@i, 
interactive program in SAIL, for DEC PDP-10 systems,
Laboratory of Statistical and Mathematical Methodology, DCRT, NIH,
Bethesda, Md.(1980). The curve fitting performed later in this
document were obtained via MLAB's Marquardt-Levenberg error
optimisation capabilities in which: (1) The standard errors are
standard deviations when the model is linear in its coefficients; (2)
Dependency values positively measure intercoefficional dependency and
uniqueness of fit, i.e., the sharpness of the stationary point; (3)
Sum of squares are a chi-squared statistic with degrees of freedom
being the number of observations less the number of coefficients; (4)
RMS error is a dimensional goodness of fit measure equal to the square
root of the chi-square/degrees-of-freedom; (5) t-statistic may be
calculated by dividing the coefficient by the standard error; (6)
r-squared may be proxied via 1-RMS-error/Best-data-value. ] 
@BEGIN(FIGURE)
@BEGIN(VERBATIM)
{EDITED FOR BREVITY}
FUNCTION FOUR(T) = A0+A2*COS((2*PI*T)/L)+A4*COS((4*PI*T)/L)+A6*COS((6*PI*T)/L)
*A0=9.1325;A4=-.1496;A2=-.1756;A6=A4/10;L=25
*FIT(A0,A2,A4,A6,L)FOUR TO M
BEGIN ITERATION 1.1, SUM OF SQUARES= 19393.1    
BEGIN ITERATION 2.1, SUM OF SQUARES= 1738.55    
BEGIN ITERATION 5.1, SUM OF SQUARES= 26.3939    
FINAL              NORMAL ERROR       DEPENDENCY
PARAMETER VALUES:  STANDARD ERRORS:   VALUES:
     4.28745         .110259@@-1      .126308    A0
    -5.05075         .143704@@-1      .257494@@-1 A2
     .464362@@-1      .153142@@-1      .127523    A4
     .832312         .146220@@-1      .223575@@-1 A6
     23.4808         .198168@@-1      .195328    L
RMS ERROR= .230681    
FINAL SUM OF SQUARES= 26.3939    
{TSTAT=389,-351,3,57,1185 (ALL SIGNIFICANT), CHISQ=.21}
@caption(Fitting of Fourier series to characteristic curve)
@END(VERBATIM)
@END(FIGURE)
@begin(figure)
@blankspace(73 mm)
@caption(Fourier-Characteristic Graph)
@end(figure)
	We can therefore conclude that the characteristic function
can be approximated as
@equation{Fourier(t)=.126+.0256cos(32.17t) +.128cos(64.33t)+.0224cos(96.5t)}
@chap(Stage Determination of Expiremental Data)
          Thereafter,the experimental data is fitted to the expected data for
the determination of scale factors. This shall permit determination of the
growth stage involved. @foot[@i] The data is fitted to the equation
@equation{f(t')=c*Fourier(a*t'+b) + [d=0],} for the determination of the scaling factors,
a and c, and the time shifting, b. It is assumed that both data have a common
ordinate origin @foot[d=0, elsewise fit  d as well].

	One should plot the Fourier function and choose what times delimits
the stage change,  t@-, t@-, and t@-, hereafter indicated
t@-=a*t'+b. And the control and analysis program can use them as
set points, @equation{ t'=(t@--b)/a}.
@begin(figure)
@begin(verbatim)
{EDITED FOR BREVITY}
*FCT FOUR(T)=A0+A2*COS(2*PI*T/L)+A4*COS(4*PI*T/L)+A6*COS(6*PI*T/L)
*L=23.4808;A0=4.28745;A2=-5.05075;A4=.0464362;A6=.832312
*A=1;B=1;C=1;FCT F(T)=C*FOUR(A*T+B);PI=ATAN(1)*4
*M=LIST(12.5,13.5,14,14.5,15,15.5,16)
*"ABOVE ARE TIME,DY OF EXPT-BELOW,ABSORBANCE"
*M COL 2=LIST(.05,.055,.055,.055,.06,.1,.12)
*FIT(A,B,C)F TO M
BEGIN ITERATION 1.1, SUM OF SQUARES= 391.777    
BEGIN ITERATION 25.6, SUM OF SQUARES= .123202@@-2 
FINAL              NORMAL ERROR       DEPENDENCY
PARAMETER VALUES:  STANDARD ERRORS:   VALUES:
    -.233554         3.40511         .999998    A
     23.0022         27.3330         .999993    B
     .819191@@-1      2.25473         .999990    C
FINAL SUM OF SQUARES= .123149@@-2 
BEGIN ITERATION 1.1, SUM OF SQUARES= .123149@@-2 
BEGIN ITERATION 1.4, SUM OF SQUARES= .123149@@-2 
FINAL              NORMAL ERROR       DEPENDENCY
PARAMETER VALUES:  STANDARD ERRORS:   VALUES:
    -.233554         3.34405         .999998    A
     23.0022         26.7837         .999993    B
     .819191@@-1      2.32294         .999990    C
CONVERGED
RMS ERROR= .175463@@-1 
FINAL SUM OF SQUARES= .123149@@-2 
*DRAW POINTS(F,-10:10:.5);DRAW M,LINETYPE 0,POINTTYPE 4;DISPLAY=T4010;DRAW POINTS(F,0:100:.3)
*DRAW 0:100&'0;DRAW 0&'0:1;DELETE AXESBOX
@end (verbatim)
@caption(Fitting of Expirement to Determine Stage)
@end(figure)
@begin(figure)
@blankspace(141 mm)
@caption(Fitted Data)
@end(figure)
@appendix(Synopsis of BioChemical Engineering)
          Yeast cells thrive best at temperatures between thirty and fourty
centigrade degrees. Water must be about seventeen twentieths pure to avoid
osmotic distortions such as crenation and bursting. While the yeast are
saccharolytic and thus saccharophagic, they need various nutrients. They
are composed of proteins, derived from amino acids. Thus potassium,
phosphorus, sulphur, magnesium @i(et seq.) must be present in the food
supply.

          Growth begins similarly to the start of an exponential curve, after
which it becomes logarithmic. The former is classified as the lag phase
during which adjustment to the new environment occurs. The latter is the
logarithmic phase, in which rapid division occurs. Then the stationary phase
begins as the population remains numerically constant and the living cells
obtain nutrients from the dead cells. This sorry state of affairs leads them
to the aptly named death or decline phase, which is approximable via a
mirrored logarithmic function.@foot{B.Faber,@I,pp.20-30,unpublished, Columbia University, Fall,1979,C1501.}

          More advanced life forms use oxygen in their metabolic reactions.
In a muscle, sugar is converted to carbon dioxide and water, via oxygen, or
aerobically. Energy is derived from this process. Electron transfer is via
the interconversion of adosine tri- and di- phosphate. The latter half of
this is named the citric acid cycle. Hexacarbonous glucose is coverted to 
pyruvic acid and follows to citrate, alpha-ketoglutarate, succinate, fumarate,
malate, oxaloacetate, acetyl, and back to citrate. Most life forms follow
this example.@foot{J.D.Watson,@i,3rd.ed.,
pp.66-81,Reading,Massachusetts:W.A.Benjamin,1976.}

          In the absence of oxygen, however, metabolism may proceed
anaerobically.  Glucose is coverted to pyruvic acid. Some bacteria, and
muscle cells, metabolise this to tricarbonous lactic acid, which results
in muscular fatigue. Yeasts, alternatively, proceed via acetaldehyde to
dicarbonous ethyl alcohol. This metabolism is not as complete, yet provides
energy.  @foot{J.J.W.Baker & G.E.Allen, @i,3rd.ed.,
pp.209-222,Reading, Massachusetts:Addison-Wesley,1977}

          In engineering work, the Monod Equation may be applied as a primary
approximation. It is of a form similar to the famed Michaelis-Menton Equation,
as well as to the Langmuir adsorption isotherm. It states that the 
growth rate, @equation
[@ovp[)]\=@ovp[)]\@-(max)c@-(i)/(K@-+c@-)|@ovp[)]\@-(max)=2@ovp[)]\|K@-(i)]
with individual nutrient concentration, @equation[c@-(i)=e@+{-@ovp<)>\t}]
@foot[J.E.Bailey & D.F.Ollis,@i,
pp.346,NY:McGrawHill,1977;J.Monod,The Growth of Bacterial Cultures,
@i:371(1949);Dr. Monod received the 1965 Nobel  Prize 
for this and other work on enzymatic adaptation, which he began during WW2.
@i O' Sullivan, D., G., Quantitative Potentialities in Enzyme
Cytochemistry. Modified Michaelis-Menten Rate Law Applicable when a
Substrate Diffuses Slowly into an Enzyme Site,@i,@b<2>,
pp.119,123,124(1962) ]

          Prof. Levenspiel restates this as
@equation{r@-=kC@-C@-/(C@-+C@-),} where c indicates cells
and A, substrate. C@- is, as above, the Monod constant, or the substrate
concentration at the semi-maximal rate. Due to inhibitory product,R, he
proposes the more generalised form,
@equation{k:=k@-=k(1-C@-/C@-@+<*>)@+.} The asterisk is indicative
of the state at which reproduction is fully inhibited. @foot[O.Levenspiel, The 
Monod Equation: A Revisit and a Generalization to Product Inhibition 
Situations, @i,pp.1671-1687(1980)]

          For a CFSTR,
@equation{t@-=(C@-+C@-)/C@-=([C/A](C@-+C@-)-C@-)/([C/A]C@--C@-)
    |C@-=0  &  kt@- @b(>)1.}
This is the equation that was developed simultaneously by Monod @foot[Monod,
Jacques, @i,@b<79>,390(1950).] and by Novick and Szilard
@foot[Novick, A., & Szilard, L., @i,@b<36>,708(1950).]

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