J. theor. Biol. (1977) 65, 633-651
Application of Quine’s Nullities to a Quantitative Organelle Pathology G. WILLIAM
MOORE?
Department of Anatomy, Wayne State University School of Medicine, Detroit, Michigan 48201, U.S.A. AND
U. N. RIEDE AND W. SANDRITTER Department of Pathology, Freiburg University, Freiburg i. Br., W. Germany (Received 16 March 1976, and in revisedform 1 June 1976) Ultrastructural morphometry makesit possibleto quantitate the pathomorphological changesin cells and cell organelles.Morphometric data gatheredon cellular reactionsto various cell injuries suggestthat the cell hasonly a limited repertoire of responseto a wide variety of injuries. As a first step in documenting this hypothesis, a self-consistentmodel of organellepathology hasbeenconstructedin termsof the generalpathology of growth disorders.This modelemploysthe nullity approach,introduced by Quine. A computer algorithm for extracting an exhaustive, pathomorphologic description from a standardized panel of morphometric measurementsis presented,and the exhaustivenessof this algorithm is proved mathematically. Over 70 panelsof morphometric data from the literature have been successfullyanalyzed using this model of organelle pathology.
1. Introduction Ultrastructural morphometry makes it possible to quantitate the pathomorphological changes in cells and cell organelles. The increasing wealth of morphometric data gathered over the past few years on cellular reactions to various cell injuries (Loud, 1968; Weibel, Staeubli, Gn%gi & Hess, 1969; Weibel, 1969; Staeubli, Hess & Weibel, 1969; Weiner, Loud, Kimberg & Spiro, 1968; Reith, 1973; Rohr & Riede, 1973) leads to the hypothesis that the cell has only a limited repertoire of response to a wide variety of injuries (Trump & Arstila, 1971; Hill, 1971; Rohr & 7 Presentaddress:Departmentof Pathology, JohnsHopkins Hospital, Baltimore, Maryland 21205,U.S.A. 633
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Riede, 1973; Chayen & Bitensky, 1973; Sandritter & Riede, 1974; Riede, 1976). One approach for documenting this hypothesis is to determine logically the different pathological states for organelles and to catalogue the observed reactions of cellular organelles in a form suitable for computer analysis. One can then determine whether the computer can pick out classes of patterns which characterize the different states of cell injury, and ascertain which of these patterns are organelle specific. 2. Methods As a first step in our program, we have to construct a self-consistent model of organelle pathology in terms of the general pathology of the growth disorders. The current model employs a subdivision of the hepatocyte (H) into seven organelle systems : nucleus (N), smooth endoplasmic reticulum (S), rough endoplasmic reticulum (R), mitochondria (M), mitochondrial cristae (C), peroxisomes (P), and lysosomes (L). These organelles are further classified as either particulate organelles (N, M, P, L) or as tubulocisternal organelles (S, R, C). These organelle systems as well as the hepatocyte as a whole are characterized by means of morphometrically determined values : volume density (V), numerical density (Z), and surface density (F). Both the pathological and morphological states of the organelles are defined in terms of combinations of these variables. Nine pathologic states are recognized: normal (N), proliferation (P), hypertrophy (T), hyperplasia (R), hypoplasia (0), atrophy (A), ageneration (E}, dysplasia (D), and dystrophy (Y). Four morphologic states are recognized: unchanged (G), vesiculated (S), vacuolated (U), and collapsed (C). These definitions are expressed in the form of mathematical implications, which are then translated for computer processing into a format reminiscent of the nullity concept, introduced by Quine (1948). The possible pathologic states for cell organelles are summarized in Table 1. 1. Proliferation (P) is defined either as a numerical increase without volume change in particulate organelles or as membrane increase (= “numerical increase of membrane units”) without volume change in tubulocisternal organelles. 2. Hypertrophy (2”) is defined as volumetric increase without change in particulate organelle number or in membrane surface of tubulocisternal organelles. 3. HyperpZasia (R) is de&red as a numerical and volumetric increase in particulate organelles and as an increase in volume and membrane surface in tubulocisternal organelles.
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TABLE 1
Summary of pathologic state definitions = increased, N = normal, 1 = decreased t Pathologic state definitions for quantitative organelle pathology
Normal = Proliferation = Hypertrophy = Hyperplasia = Hypoplasia = Atrophy = Ageneration = Dysplasia = Dystrophy =
N P T R 0 A E D Y
t Nuclei, mitochondria,
Volume
Numerical
density = V for all organelles
density = Z for particulate organellest
Surface density = F for tubulocisternal organellest
N
peroxisomes, lysosomes and hepatocytes.
$ Smooth and rough endoplasmic reticulum and mitochondrial cristae.
4. Hypoplasia (0) is the opposite of hyperplasia. It is defined as a numerical and volumetric decrease in particulate organelles and as a volume and membrane surface decrease in tubulocisternal organelles. 5. Atrophy (A) is the opposite of hypertrophy. It is defined as volumetric decrease in organelles without change in number of particulate organelles or in membrane surface of tubulocisternal organelles. 6. Ageneration (E) is the opposite of proliferation. It is characterized as a numerical reduction in particulate organelles or as a reduction in membrane surface in tubulocisternal organelles without simultaneous volumetric change of the organelles. 7. Dysplasia (0) corresponds to a numerical reduction with simultaneous volumetric increase in particulate organelles, or to a membrane surface reduction with simultaneous volumetric increase in tubulocisternal organelles. 8. Dystrophy (Y) is the opposite of dysplasia. It is defined as numerical increase and volume decrease in particulate organelles, and as membrane surface increase and volume decrease in tubulocisternal organelles. These definitions are constructed according to a master plan: all changes designated with “trophy” correspond to volume changes in the organelle. All changes with “plasia” show volume and numeric changes in particulate organelles or volume and membrane surface changes in tubulocisternal organelles. Ageneration and proliferation describe disturbances in organelle
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neogenesis based upon numeric changes in particulate organelles or surface changes in tubulocisternal organelles. Morphometric parameters (volume density = Vvnn, numerical density and surface density = S,& for the nucleus and for the hepatocyte = NVNH, have unit volume of liver tissue (cm”) as a reference system, and correspond to the morphometric symbols VVNH, NVNH, SVNH (Weibel, 1969). The morphometric parameters for all cytoplasmic organelles have volume density of cytoplasm (Vvc) as a reference system, and correspond to the morphometric symbols V&Vv,-, NvrJVvc, SvMo/Vvc (Rohr 8z Riede, 1973). Volumetric increase in a given organelle may be observed in the cytoplasm even in the absence of absolute volume change, as in the case of a decreasing volume proportion of hepatocyte per unit volume liver tissue. The same holds for liver cells in the case of an alteration in total organ weight (= total organ volume). This is taken into account in our implications. The morphoZogicaZ state of the tubulocisternal organelles (S, R, C) may be characterized additionally in terms of specific membrane surface. This parameter corresponds to surface density per volume density of a given organelle, and, say, for smooth endoplasmic reticulum, is denoted by the morphometric symbol, SVSE,JVvSER (Weibel, 1969; Rohr & Riede, 1973). The four possible morphologic states for these tubulocisternal organelles are : unchanged (normal appearance, G), vesiculated, vacuolated, or collapsed. The vesiculated state (S) is defined as disproportionate changes in the volume density and surface density of an organelle resulting in a decrease in its specific surface. In the case of the smooth endoplasmic reticulum, vesiculation of the structural elements is always present. The vacuolated state (U) is defined as opposite behavior in the surface density and volume density in which the organelle volume increases and the organelle surface along with the specific surface of the organelle decreases. The tubulocisternal organelles invariably appear swollen in this state. The collapsed state (C) denotes an increase in specific surface of the organelle. In this case, the tubuli and cisternae of the smooth or rough endoplasmic reticulum are narrowed, whereas in mitochondria we observe shorter distances between the cristae.
3. Conditions A condition is a statement which is either true or false. Direct measurements and derived conclusions must be expressed in the form of conditions before they are suitable for handling in our system. Pathologic and morphologic states for a given organelle are designated in organelle mnemonic
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(N, S, R, M, C, P, L, or H) first, followed by the pathologic state mnemonic (N, P, T, R, 0, A, E, D, Y) or morphologic state mnemonic (G, S, C, U), respectively. For example MT denotes the mitochondria are hypertrophied and RS denotes the rough endoplasmic reticulum is vesiculated. Measurements require an additional symbol denoting quantity: J = significantly decreased, - = normal, 1 = significantly elevated. For example, SVT denotes “the smooth endoplasmic reticulum volume per unit volume cytoplasm is significantly elevated”. The negation of a condition is given by
TABLE 2
Mnemonic letters for quantitative organelle pathology Z = V = F = Q= G = S= C = U = W = 1= 2= 4=
numerical density volume density surface density specific membrane surface unchanged morphology vesiculated collapsed vacuolated liver weight oligoploid normoploid polyploid
N P R T 0 A E D Y
= = = = = = = = =
normal status proliferation hyperplasia hypertrophy hyperplasia atrophy ageneration dysplasia dystrophy
prefacing the condition with a minus sign, -. Thus, -MT denotes the mitochondria are not hypertrophied, -RS denotes the rough endoplasmic reticufum is not vesiculated, and -SVt denotes the smooth endoplasmic reticulum volume per unit volume cytoplasm is not significantly elevated. Each condition is designated by a unique Quine number (as opposed to numbers which designate amounts). Table 3 shows the Quine number for each condition in our system. For example, MT = 145, RS = 052, and svt = 013. 4. Implications
All statements which are processed by our system are initially formulated as implications. Each implication is either absolute or relative. An absolute implication corresponds to a single condition which is always true. For example, the statement “it is always true that A” would be expressed as ==-A. A relative implication is a string of two or more conditions divided by *.
0
042
005
0
0
030 031 032
029
012 015 016 021 022
011
006
0
0
033 034
0
023 024
018
017
0
0
Q
0
0
055
0
051
047
0
0
0
0
056
0
052
048
0
0
0
0
057
0
053
049
0
0
0
0
058
0
054
050
0
0
Morphology GSCUNPRTOAEDY
118
117
116
115
114
113
112
111
128
127
126
125
124
123
122
121
138
137
136
135
134
133
132
131
148
147
146
145
144
143
state
158
157
156
155
154
153
168
167
166
165
164
163
162
151 161
142 152
141
Pathological
Oligoploid = 043. Polyploid = 044. Liver Weight: f = 045, j, = 046. 0 = no measurements. In the morphometry section, the upper Qume number corresponds to f, the lower Quine number to 4.
040
039 041
037 038
0
020 027 028
Lysosomes = L
0
02.5 026
019
035 036
= M
0
013 014
001 003 004 009 008 010 002 007
Morphometry V F
Perosisomes = P
Cristae = C
Chondrioma
Rough ER = R
Smooth ER = S
Hepatocyte = H
Nucleus = N
Z
Quine numbers for each condition
TABLE 3
178
177
176
175
174
173
172
171
188
187
186
185
184
183
182
181
198
197
196
195
194
193
192
191
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For example, A 3 B which means, “whenever A is true, then B is true”. Relative implications can be compounded to desired complexity. For example, A*BvC A*B&C
(9 (ii)
A & B * (C & D) v (E & F)
(iii)
(A * B) * (C v D) & (E * F)
(iv)
The left side of the major * sign is called the hypothesis; the right side is called the conclusion. “Logical and” is denoted “W; “logical or” is denoted “v”. Eventually, these implications must be expressed in standard form. A relative implication is said to be expressed in standard form when the hypothesis contains exactly one condition, and all conditions in the conclusion are separated by logical ors. Statement (i) is already in standard form. Statement (ii) expands into two statements in standard form: A+-B and A * C. Statement (iii) becomes A*
-BvCvE A* -BvCvF A=> -BvDvE and A=+ -BvDvF. Statement (iv) becomes -A+CvD B*CvD --A* -EvF and B*
-EvF.
5. Nullities
In his Theory of Deduction, Quine (1948) presents the concept of nullity. A nullity is a statement of what does not exist (or is not true). Quine uses the symbol “y” to preface each nullity, but we shall dispense with it without loss
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of clarity in this paper. Qume states : “Let us write ‘yl? to mean ‘there are no F’...‘ - yF’ denies that there are no F, and thus affirms that there are F.” Similar devices for dealing specifically with biomedical data are discussed by Feinstein (1967) and Ledley (1965). The relative implication 1 =+ 2 means that conditions 1 and -2 cannot coexist. The nullity signifying “nonco-existence of 1 and -2” is expressed in shorthand as 1 -2. Similarly, the implication 1 * 2 v 3 is equivalent to the nullity 1 -2 -3. An absolute implication, say => 1, is equivalent to the nullity of opposite sign: - 1. In other words, “it is always true that 1” is equivalent to, the “nonexistence of - 1”. All implications written in standard form are converted to nullities in this fashion. Any collection of nullities is called a nullity set. The size of a nullity is the number of explicitly state (nonzero) conditions in the nullity. A nullity of size one is called a monoconditional. A nullity of size two is called a biconditional. A nullity of size k is called a k-conditional. There is also a nullity of size zero, which will be discussed further below. It is called the “o-nullity” or simply “w” because its presence in the system terminates all calculations and effectively spoils the sense of the system. Extensive effort may be required to establish the absence of o in a given system. In a system into which no data have been introduced, a monoconditional is undesirable, since it indicates the presence of a condition which is predetermined, without reference to data. This doesn’t formally spoil the system, but it is a warning suggestive of poor input logic. Biconditionals one of whose nonzero members is a label (experiment number) is a device both for introducing data and extracting conclusions relevant to specific experiments. 6. Null Addition
The fundamental operation which we perform upon pairs of nullities is called null-addition, denoted CD.The importance of this operation will be discussed below. First, the conditions in each nullity are lined up in ascending order of absolute value, with either zeroes or blanks representing the empty columns in the nullity. For example: 1 2 3 2
3 4
are properly lined up. Null addition is possible only if there is exactly one column in which one nullity has the positive element and the other nullity has the negative element (“sign reversal”). For example, null-addition is not possible on the above nullity pair, because no column exhibits sign reversal.
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On the other hand, this nullity pair 1 -2
3 2
-3
4
is unsuitable for null-addition because more than one column (second and third) have sign reversal. This nullity pair 1 -2 2
3 3 4
can be null-added. The formula for null addition is 0 a a a
@ o=o 68 a=a 8 -a = 0 o 0=0 63 a=a
(a # 0).
Thus, 1 -2 8
3
2
3 4 3 4
1
The nullity, 1 3 4, is called the null-sum. The special case in which the oaly nonzero column present in both nullities is the sign reversal column results in a null-sum of “w”. For example, -7 8
7
The significance of w is that its presence signals a contradiction within the system. We cannot be certain that w is absent from our system until we have performed exhaustive null additions, to obtain the canonical form. For economic reasons, we employ two shortcuts in our computer algorithm : we retain only (i) null-sums which contain an experiment label and (ii) nullsums which contain no more members than the larger-sized of the two nullities which are being summed. Both shortcuts have mathematical justification (Moore et al., manuscript in preparation). For label x (i.e., experiment number x), the former shortcut guarantees that all label-speczjk conclusions (but not necessarily all conclusions) will be found. A conclusion is said to be label-specific if it is true for label x, but not true for some conceivable experiment whose data are inconsistent with label x. All the conclusions which interest us in this model for quantitative organelle pathology (save for the o-nullity, which was ruled out separately) are T.B. 42
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label-specific. The second shortcut is valid when all label-specific nullities turn out to be biconditional or monoconditional. This requirement was satisfied by all our data sets. 7. Example Tables 4-6 present a worked out example of the nullity method applied to morphometric data. Experiment 928 involved rats with oncocytic transformation of the cytoplasm (Riede et a/., 1975a). Table 4 shows the actual TABLE 4 Example of morphometric analysis on rats with oncocytic transformation of the cytoplasm? (Experiment 928) Mnemonic
Symbols Quine Nos
NZT
001
NVJ NF+ HZ+ HV-
J-9 SVJ SFw
007 -009
-010 011 014
-015
LZJ LVff
040 -041
Wj,
-042 046 043
RQt MZt MV? MW CW
CQt PZJ.
Oligoploid
-
005
PVJ
RVJ RF’?
Control rats
0.23 0.049 6.2
0.58 0.026 6.3
m2/cm3 cm3/cm3 m2/cm8
31.7 0.07 3.2 35.2 320~10~ 0.22 2.06 4.7 15.7 145x log
91.2 0.02 6.7 125.5 525x lo9 0.53 15.2 29.1 23.5 65 x lo9 om9 20x109 oaO5
m2/cm3 cm3/cm3 m “/cm 3 m2/cm3 cme3 cm3/cm3 m2/cm3 ma/cm3 ma/cm3 cmm3 cm3/cm3 cme3 cm3/cm3
0.015
liver weight of diploid: tetraploid
Dimension
0.091 278~10~ 0.80
45 x 109 0.005
I:atio
Rats with oncocytic transfonnationf
cmF3 cm3/cm3 m2/cm3 cm-” cm3/cm3
162x lo6 0.052 0.037 160~10~ 0.85
004
-016 017 020 021 023 025 027 029 031 033 036 038
SQt
Morphometry
7.5 g 60:40
345x106
0444
3.og 95:5
nuclei
t Riede et al., 197%. $ Mean values from five animals each. S.E. < 10% of the mean. Symbols for the morphometric analysis described by Weibel et al., 1968; Weibel, 1969; Rohr & Riede, 1973.
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data for this experiment and the appropriate control values. For example, the volume, surface, and specific surface is given for the smooth endoplasmic reticulum. Volume is significantly decreased (control = O-049, experimental = O-026), which corresponds to the mnemonic condition SVJ and the Quine number 014 (Table 3). Specific surface is significantly elevated (control = 31.7, experimental = 91*2), which corresponds to the mnemonic condition SQt and the Quine number 017. Surface is essentially normal (control = 6.2, experimental = 6.3). Rather than construct a separate Quine number for SF-, we observe that SF- is the same as stating that - SFf and - SFJ (normal is equivalent to not-elevated and not-decreased). Thus experiment 928 gives rise to the following relative implications (by way of actual measurement):928 * 014and928 =P -015and928 * -016and928 =+ 017. In nullity form, we have 928 -014 and 928 015 and 928 016 and 928 -017. The complete set of input nullities for experiment 928 (i.e., based upon actual measurements) is summarized in Table 5(a).
TABLE
5
(a) Input nullities for experiment 928. (b) Output nullities for experiment 928 (a> 928 928 928 928 928 928 928 928 928 928 928 928 928
-001 -004 -005 -007 E -011
-014 015 016 -017 -020 -021
0.2
__928 928 928 928 928 928 928 928 928 928 928 928 928
-023 -025 -027 -029
-031 -033 -036 -038 -040 041
042 -046 -043
928 928 928 928 928 928 928 928 928 928 928
-049
-053 -057 -191
-162 -163 -194
-135 -136 -157 -178
The input nullities in Table 5(a) are entered into program CANON, which works out the canonical form as described above. The conclusions for experiment 928 are summarized in Table 5(b). For example, the nullity 928 -049 corresponds to the following relative implication: 928 =z. 049. In mnemonic notation, 928 => SC; in other words, “the smooth endoplasmic reticulum is collapsed in experiment 928”. The data and conclusions for experiment 928 are summarized in Table 6.
Nucleus= N Hepatocyte= H SmoothER = S RoughER = R Chrondrioma= M Cristae= C Peroxisomes = P Lysosomes = L Ploidy (1, 2, 4) L. weight= W
Morphometric Morphology
ZVFQGSCUNPRTOAEDY
Morphometry
data and conclusions for experiment 928. Description: cytoplasm. (Riede et. al., 1975)
TABLE 6
Pathological
state
oncocytic transformation
of
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8. Results Computer analysis shows that every definition in our system is nonvacuous (i.e., there exist some conditions under which the definition is satisfied). Within each organelle, every pair of definitions is mutually exclusive with complete data. We have analyzed data from liver parenchymal cells undergoing five different metabolic disturbances: inhibition of protein synthesis by cycloheximide (Riede, Sebass & Rohr, 1971), hypothyroidism (Riede et al., 1975b), exsiccosis (Riede, Kreutzer, Robausch, Kiefer & Sandritter, 1974), vitamin E deficiency (Riede et al., 1971), and oncocytic transformation in the cytoplasm (Riede et al., 1975a). Our analysis shows that it is possible to specify the pathologic states for all organelles and the hepatocyte as a whole, as well as the morphologic states for the tubulocisternal organelles (rough and smooth endoplasmic reticulum, mitochondrial cristae). For each of the seven organelles in each of the five experiments, there exists one and only one pathologic state; for each of the tubulocisternal organelles in each experiment, there exists one and only one morphologic state. 9. Discussion The introduction of definitions from general pathology into quantitative organelle pathology is important, but requires that we employ precise terminology. Our investigations with a model of organelle pathology suggest that it is possible to predict the morphological and pathological status of a cell largely on the basis of combinations of a few measurements. In other words, these morphometric measurements inherently contain virtually all the information we expect to gain from a simple description of cell and organelie status. Since the morphometric measurements are objective data and the computer logic is the same for every experiment, we have the added security that we are not prejudicing the interpretation of any particular experiment in terms of subjective expectations. Of course the computer logic itself is based upon certain expectations, but this logic is at least completely specified and internally consistent. The concept of proliferation @roles ferre-bear offspring) can be developed both for particulate organelles and for tubulocisternal organelles (endoplasmic reticulum and mitochondrial cristae). Proliferation at the organelle pathologic level is seen as a simple increase in organelle number or membrane surface (= numerical increase in membrane units), and often accompanies the early phase of sublethal cell damage (Riede et al., 1971; Rohr & Riede, 1973; Chayen & Bitensky, 1973). In organ pathology, hypertrophy refers to an increase in cell size along with changes which increase the size of the organ
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(Goss, 1966; Robbins, 1974; Walter & Israel, 1974). This concept is readily carried over into quantitative organelle pathology. The same holds for hyperplasia, which is defined in cellular pathology (Virchow, 1858) as an increase in the cell number in an organ or tissue with simultaneous volume increase (Goss, 1966; Richter, 1974; Robbins, 1974; Walter & Israel, 1974). The concept of atrophy is used to describe conditions under which a shrinkage of cell size follows loss of the cellular substance (Robbins, 1974; Walter & Israel, 1974). This also carries over to the realm of quantitative organelle pathology. In quantitative organelle pathology, we characterize “ageneration” as a numerical or surface reduction of an organelle without simultaneous volume changes. Organelle neogenesis stops, and we see either abnormally large organelles or abnormally dilated compartments (cf. Riede et aZ., 1971, 1975b). Concepts of dystrophy and dysplasia in organ pathology vary from author to author (Walter & Israel, 1974). In quantitative organelle pathology, we characterize these concepts as opposite changes (one increasing, the other decreasing) in the volume versus the number, or in the volume versus the membrane surface, of the organelle. We can characterize all possible changes in quantitative organelle pathology with these eight concepts in addition to the normal state. As shown by our computer analysis, the individual pathologic conditions are defined by logically nonvacuous implications. Evaluation of over 70 morphometric analyses on liver cells undergoing various metabolic disturbances and cell damage conditions (Riede & Moore, in preparation) has shown furthermore that our nine concepts for the pathologic state of the organelle have sufficiently sharp definitions that no overlap is possible between concepts. The assignment of morphometric data to various morphologic states is very time-consuming in the absence of a computer, and subject to many unconscious errors. Thus computerization disciplines our thought processes about organelle pathology, and provides us with a convenient tool for applying these insights to a large mass of data. Finally, the existence of a collection of logically consistent definitions and implications allows us to address deeper questions in organelle pathology: what reaction patterns are detectible over large bodies of experimental data? An approach to this question is the subject of our next paper. 10. The Mathematical
Argument
We can sum up the mathematical approach to our model by saying that we think about our problems with implications, but we “think about thinking” with truth tables. A truth table is a set of entries where each entry expresses a true-false status for each condition in the system. In a maximal truth table,
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every possible entry is present. For example, the maximal truth table for three conditions (numbered 1, 2, 3) would have eight entries, as follows: 1 2 2 1 1 -2 1 -2 -1 2 2 -1 -1 -2 -1 -2
3 -3 3 -3 3 -3 3 -3.
The maximal truth table for IZ conditions would have 2” entries. An implication, such as 1 =S 2, is equivalent to the nullity, 1 -2. In the truth table world, this means that all entries in the initial truth table which simultaneously containing 1 and -2 must be deleted. In the above truth table, this means deleting lines 1 -2
3
and 1 -2 If we also have the implication 2 3), then we must delete lines
-3.
3 =z=-2 in our system (equivalent nullity, 1
2
3
2
3.
and -1 By the rule of null-addition, follows :
we can combine nullities
I -2 and 2 3 as
1 -2 es 2 3 1 3’ This new nullity,
1 3, gives us the right to delete lines 1
2
3
and 1 -2
3
from the truth table. But of course we have already done so. In other words, null-addition merely generates tautologous nullities. However, these new
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nullities may provide us with insights we might otherwise not have had, as we shall see below. The reason we must resort to mathematics is that while the truth table is an indisputably complete description of any diagnostic system, it is also very cumbersome. Thus we shall develop computationally complete algorithms in the nullity world, then justify them in the truth table world. In studying a system containing n conditions, it is convenient to Quinenumber the conditions consecutively from 1 to n. Instead of the usual representation, we denote each nullity with an n-vector containing only Is, OS, and - 1s. Each positive condition, i, in a given nullity appears as a 1 in the ith column of the corresponding n-vector. Each negative condition, -i, in a given nullity appears as a - 1 in the ith column of the corresponding n-vector. All other colunms in the n-vector are zero. The zero-vector is known as w. Null addition proceeds as before. For example, in a seven-condition system, the null addition 1 -2 3 8 2 3 4 1 3 4 would be represented n-vectorially
as
(1, - 1, 1, 0, 0, 030) Q (0, 1, 1, LO, 090) (1, 0, 1, l,O, O,O) Entries in the truth table are similarly represented. For example, the truth table entry 1 -2 3 4 - 5 6 - 7 would be represented n-vectorially as (1, -1, 1, 1, -1, 1, - 1). We denote the maximal truth table (no entries deleted) as X. If r and s are two n-vectors, then we say that Y covers s, denoted Y < s, if for every ri # 0, 1 I i I n, si = ri. The relationship between a nullity set, S, and the truth table, T, which is represented by S, is that t E T if and only if there exists no s E S such that s covers t. We denote the statement S represents T by writing r(S) = T. A variety of different nullity sets may represent the same truth table, T. It is desirable to find a representation for T which is in some sense more compact than any other nullity set. For one thing, if distinct r, s E S and r I s, then s is clearly a redundant member of S. In other words, for any t E X-T such that s I t, r I t as well. Taking this notion a bit further, let us imagine the maximal set R such that r(R) = r(S). Now remove all redundant elements in R. This new set is called the canonical form of S,
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denoted /ISI/. Since this set IIS]I is unique, we say that ]Is// = r-r(T). The burden of our mathematical argument will be to demonstrate that IIS/ results from progressively expanding S by means of null-adding all possible pairs in S to exhaustion, with removal of redundant elements as they appear. The definition in our system are summarized below: Definition 1. (Nullity set). S is a nullity set if and only if for every s E S, s = (sl, . . . , s,) and xi = -1, 0, or 1, for every i, 1 I i < n. Definition 2. (Truth table). T is a truth table if and only if for every t E T, t = (t1, . . . , t,) and ti = -t 1 for every i, 1 I i I n. The maximal T is
denoted X. X is akin to the unit cube in graph theory (Deo, 1974). DeJinition 3. (Null addition). Let q, r E S- (0). Then s = q o YE S if and only if, (i) there exists a unique j, 1 I j I n, such that qj = -rj # 0 and sj = 0, (ii) for every i, 1 I i # j I n, such that qi # 0, si = qi, and (iii) for every i, 1 I i # j I n, such that ri # 0, si = ri.
4. (Cover). Let r, s E S. Then r I s if and only if for every = ri.
Definition
Yi # 0,
Si
Definition 5. (Representation).
if and only ifs I Definition
Let s E S. Then r(s) = T c X when t E X- T
t.
For nullity set S,
6. (Representation). r(s)
=
n S&S
r(s)
Dejnition 7. (Redundant element). For nullity redundant element in S if r I s and r # s.
set S with r, s E S, s is a
Definition 8. (Canonical form). For nullity set S, the canonical form for S, denoted IIsII, is that set such that if R is the maximal set for which r(R) = r(S) and Q is the set of redundant elements in R, IIsll = R- Q.
Our theorem shows that the canonical form for S is obtained of successive null-additions and redundant element removals.
by
a process
Theorem. (Canonical form). For nullity set S, let r(S) = T and R be the maximal set such that r(R) = T. (i) Let P be obtained from S by any sequence of null-additions and redundancy removals. Then r(P) = T. (ii) Let r E R. Then for each p 2 r there exists a P obtained from S by null-additions and redundancy removals such that p E P.
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Proof. Part I. Without loss of generality, let P be the first set obtained from S by a sequence of null-additions and redundancy removals such that r(P) # T, and let Q be the immediately preceding set [where r(Q) = T]. Obviously a redundancy removal could not change r(Q). Consider the q, r E Q such that r(Q u {q @r>) # T. There must exist a t E T such that s = q Q r I t. By Definition 3(i) of null addition and without loss of generality, there exists a uniquej, 1 5 j I ~1,such that qj = 1 and rj = - 1. Without loss of generality, let tj = 1. Since q $ t, by Definition 4 of cover, there exists an i, I I i # j < iz, such that 0 # qi = - ti. By Definition 3(ii) of null-addition, si = qi = - ti. By Definition 4 of cover, s $ t. Contradiction. Part II. Without loss of generality, order the n conditions such that all conditions i at which ri = 0 precede all conditions j at which rj # 0. Consider the first h at which there exists t 2 Y, u 2 r such that t, = 2kh# 0, ti = Ui = 0 for i < h, and ti = Ui # 0 for i > h and no p E P obtained after exhaustive null-additions such that p < t and p I U. Consider the distinct q, s E P such that q I t and s I u. Then qi = Si = 0 for i < h, qh = t, = -u, = -s, (or else a p E P would exist), and qi = ti if qi # 0 and si = ti if si # 0 for i > h. By Definition 3 of null-addition, p = q Q s exists and satisfies the above properties. Contradiction. This work was supported in part by a National ScienceFoundation grant to ProfessorM. Goodman and in part by the DeutscheForschungsgemeinschaft. We wish to thank Miss Elaine Krobock and Miss Karin Bensing for secretarial assistance. REFERENCES BOLENDER, R. (1974).Fedn. Proc. Fed. Am. SOCK exp. Biol. 33,2187. CHAYEN, J. & BITENSKY, L. (1973).In Cell BioIogy in Medicine (E.
E. Bittar, ed.), pp. 595-680.New York-London-Sydney-Toronto: JohnWiley & Sons.
DEO,
N. (1974). Graph l7zeory with Applications
to Engineering
and Computer Science,
pp. 348354.EnglewoodCliffs, N.J.: Prentice-Hall,Inc. FEINSTEIN, A. R. (1967).CZinfcdZJudgment, pp. 156-228.Baltimore,Md.: Williams& Wilkins. GOSS, R. J. (1966).Science, N.Y. 153,1615. HILL, R. B. (1971).In Principles ofPathobioZogy (M. F. LaVia & R. B. Hill, eds),pp. 3-8. NewYork, London,Toronto: Oxford UniversityPress. JEZEQUEL, A. M., KOCH, M. & ORLANDI, F. (1974).J. Br. Sot. Gastroenterology Gut 15,737. LEDLEY, R. S. (1965).Use of Computers in Biology and Medicine, pp. 770-819. New York: McGraw-Hill Book Company. LOUD,A. V. (1968).J. Cell. Biol. 37,27. MAGALHAES, M. M. 62 MAGALHAES, M. C. (1970).J. Ultrastruct. Res. 32, 32. QUINE,W. V. (1948).Theory of Deduction, Parts Z-ZV, pp. 54-81. Cambridge,Mass.: Harvard CooperativeSociety.
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REITH, A. (1973). Lab. Invest. 29, 216. RICHTER, G. W. (1974). Circ. Res. 34, H-27. RIEDE, U. N., SEBASS, C. & ROHR, H. P. (1971). Virchows Arch. Abt. B. Zellpath. 9, 16. RIEDE, U. N., SENN, E. & ROHE, H. P. (1972). Cytobiology 5,181. RIEDE, U. N., STITNY, C., ALXHAUS, S. & ROHR, H. P. (1972). Beitr. Path. 145, 24. RBDE, U. N., KREUTZER, W., ROBAUSCH, TH., KIEFER, G. & SANDRITTER, W. (1974). Beitr. Path. 153, 379. RIEDE, U. N., SCHMITZ, E., ROBAUSCH, TH., KIEFER, G., GR~~NHOLZ, D. & SANDRITTER, W. (1975a). Beitr. Path. 154, 140. RIEDE, U. N., SCHMIDT, E., KIEFER, G., ROHRBACH, R. & SANDR~ER, W. (1975b). Beitr. Path. 155,263. RIEDE, U. N. (1976). In PuthobioZogy of Cell Membranes, Vol. ZZZ(B. F. Trump & A. U.
Arstila, eds). New York: Academic Press. S. L. (1974). Pathologic Basis of Disease, pp. 11-19. Philadelphia-LondonToronto: W. S. Saunders Co. ROHR, H. P. & -DE, U. N. (1973). Cur-r. Top. Path. 58, 1. SANDRI~ER, W., RIEDE, U. N. (1974). In Lehrbuch der allgemeinen Pathologic (W. Sandritter & G. Beneke, eds), pp. 191-226. Stuttgart: Schattauer Vet-lag. STAEUBLI, W., HESS, R. & WEIBEL, E. R. (1969). J. Cell BioZ. 42, 92. TRUMP, B. F. & ARSTILA, A. (1971). In Principles of Puthobiology (M. F. LaVia & R. B. Hill, eds), pp. 9-95. New York-London-Toronto: Oxford University Press. VIRCHOW, R. (1858). Die Cellularpathologie in ihrer Begriindung auf physiologische und pathologische Gewebelehr, pp. 58-60. Berlin: A. Hirschwald Verlag. WALTER, J. B. & ISRAEL, M. S. (1974). General Pathology, pp. 297-311. Edinburgh and London : Churchill Livingstone. WEIBEL, E. R. (1969). Znt. Rev. CytoZ. 26, 235. WEIBEL, E. R., STAEUBLI, W., GNXGI, H. R. & HESS, F. A. (1969). J. Cell BioZ. 42, 68. ROBBINS,
WIENER,
J., LOUD,
A. V., &BERG,
D. V. & SPIRO, D. (1968).
J. Cell BioZ. 37, 47.
Application of Quine’s Nullities to a Quantitative Organelle Pathology G. WILLIAM
MOORE?
Department of Anatomy, Wayne State University School of Medicine, Detroit, Michigan 48201, U.S.A. AND
U. N. RIEDE AND W. SANDRITTER Department of Pathology, Freiburg University, Freiburg i. Br., W. Germany (Received 16 March 1976, and in revisedform 1 June 1976) Ultrastructural morphometry makesit possibleto quantitate the pathomorphological changesin cells and cell organelles.Morphometric data gatheredon cellular reactionsto various cell injuries suggestthat the cell hasonly a limited repertoire of responseto a wide variety of injuries. As a first step in documenting this hypothesis, a self-consistentmodel of organellepathology hasbeenconstructedin termsof the generalpathology of growth disorders.This modelemploysthe nullity approach,introduced by Quine. A computer algorithm for extracting an exhaustive, pathomorphologic description from a standardized panel of morphometric measurementsis presented,and the exhaustivenessof this algorithm is proved mathematically. Over 70 panelsof morphometric data from the literature have been successfullyanalyzed using this model of organelle pathology.
1. Introduction Ultrastructural morphometry makes it possible to quantitate the pathomorphological changes in cells and cell organelles. The increasing wealth of morphometric data gathered over the past few years on cellular reactions to various cell injuries (Loud, 1968; Weibel, Staeubli, Gn%gi & Hess, 1969; Weibel, 1969; Staeubli, Hess & Weibel, 1969; Weiner, Loud, Kimberg & Spiro, 1968; Reith, 1973; Rohr & Riede, 1973) leads to the hypothesis that the cell has only a limited repertoire of response to a wide variety of injuries (Trump & Arstila, 1971; Hill, 1971; Rohr & 7 Presentaddress:Departmentof Pathology, JohnsHopkins Hospital, Baltimore, Maryland 21205,U.S.A. 633
634
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Riede, 1973; Chayen & Bitensky, 1973; Sandritter & Riede, 1974; Riede, 1976). One approach for documenting this hypothesis is to determine logically the different pathological states for organelles and to catalogue the observed reactions of cellular organelles in a form suitable for computer analysis. One can then determine whether the computer can pick out classes of patterns which characterize the different states of cell injury, and ascertain which of these patterns are organelle specific. 2. Methods As a first step in our program, we have to construct a self-consistent model of organelle pathology in terms of the general pathology of the growth disorders. The current model employs a subdivision of the hepatocyte (H) into seven organelle systems : nucleus (N), smooth endoplasmic reticulum (S), rough endoplasmic reticulum (R), mitochondria (M), mitochondrial cristae (C), peroxisomes (P), and lysosomes (L). These organelles are further classified as either particulate organelles (N, M, P, L) or as tubulocisternal organelles (S, R, C). These organelle systems as well as the hepatocyte as a whole are characterized by means of morphometrically determined values : volume density (V), numerical density (Z), and surface density (F). Both the pathological and morphological states of the organelles are defined in terms of combinations of these variables. Nine pathologic states are recognized: normal (N), proliferation (P), hypertrophy (T), hyperplasia (R), hypoplasia (0), atrophy (A), ageneration (E}, dysplasia (D), and dystrophy (Y). Four morphologic states are recognized: unchanged (G), vesiculated (S), vacuolated (U), and collapsed (C). These definitions are expressed in the form of mathematical implications, which are then translated for computer processing into a format reminiscent of the nullity concept, introduced by Quine (1948). The possible pathologic states for cell organelles are summarized in Table 1. 1. Proliferation (P) is defined either as a numerical increase without volume change in particulate organelles or as membrane increase (= “numerical increase of membrane units”) without volume change in tubulocisternal organelles. 2. Hypertrophy (2”) is defined as volumetric increase without change in particulate organelle number or in membrane surface of tubulocisternal organelles. 3. HyperpZasia (R) is de&red as a numerical and volumetric increase in particulate organelles and as an increase in volume and membrane surface in tubulocisternal organelles.
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TABLE 1
Summary of pathologic state definitions = increased, N = normal, 1 = decreased t Pathologic state definitions for quantitative organelle pathology
Normal = Proliferation = Hypertrophy = Hyperplasia = Hypoplasia = Atrophy = Ageneration = Dysplasia = Dystrophy =
N P T R 0 A E D Y
t Nuclei, mitochondria,
Volume
Numerical
density = V for all organelles
density = Z for particulate organellest
Surface density = F for tubulocisternal organellest
N
peroxisomes, lysosomes and hepatocytes.
$ Smooth and rough endoplasmic reticulum and mitochondrial cristae.
4. Hypoplasia (0) is the opposite of hyperplasia. It is defined as a numerical and volumetric decrease in particulate organelles and as a volume and membrane surface decrease in tubulocisternal organelles. 5. Atrophy (A) is the opposite of hypertrophy. It is defined as volumetric decrease in organelles without change in number of particulate organelles or in membrane surface of tubulocisternal organelles. 6. Ageneration (E) is the opposite of proliferation. It is characterized as a numerical reduction in particulate organelles or as a reduction in membrane surface in tubulocisternal organelles without simultaneous volumetric change of the organelles. 7. Dysplasia (0) corresponds to a numerical reduction with simultaneous volumetric increase in particulate organelles, or to a membrane surface reduction with simultaneous volumetric increase in tubulocisternal organelles. 8. Dystrophy (Y) is the opposite of dysplasia. It is defined as numerical increase and volume decrease in particulate organelles, and as membrane surface increase and volume decrease in tubulocisternal organelles. These definitions are constructed according to a master plan: all changes designated with “trophy” correspond to volume changes in the organelle. All changes with “plasia” show volume and numeric changes in particulate organelles or volume and membrane surface changes in tubulocisternal organelles. Ageneration and proliferation describe disturbances in organelle
636
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neogenesis based upon numeric changes in particulate organelles or surface changes in tubulocisternal organelles. Morphometric parameters (volume density = Vvnn, numerical density and surface density = S,& for the nucleus and for the hepatocyte = NVNH, have unit volume of liver tissue (cm”) as a reference system, and correspond to the morphometric symbols VVNH, NVNH, SVNH (Weibel, 1969). The morphometric parameters for all cytoplasmic organelles have volume density of cytoplasm (Vvc) as a reference system, and correspond to the morphometric symbols V&Vv,-, NvrJVvc, SvMo/Vvc (Rohr 8z Riede, 1973). Volumetric increase in a given organelle may be observed in the cytoplasm even in the absence of absolute volume change, as in the case of a decreasing volume proportion of hepatocyte per unit volume liver tissue. The same holds for liver cells in the case of an alteration in total organ weight (= total organ volume). This is taken into account in our implications. The morphoZogicaZ state of the tubulocisternal organelles (S, R, C) may be characterized additionally in terms of specific membrane surface. This parameter corresponds to surface density per volume density of a given organelle, and, say, for smooth endoplasmic reticulum, is denoted by the morphometric symbol, SVSE,JVvSER (Weibel, 1969; Rohr & Riede, 1973). The four possible morphologic states for these tubulocisternal organelles are : unchanged (normal appearance, G), vesiculated, vacuolated, or collapsed. The vesiculated state (S) is defined as disproportionate changes in the volume density and surface density of an organelle resulting in a decrease in its specific surface. In the case of the smooth endoplasmic reticulum, vesiculation of the structural elements is always present. The vacuolated state (U) is defined as opposite behavior in the surface density and volume density in which the organelle volume increases and the organelle surface along with the specific surface of the organelle decreases. The tubulocisternal organelles invariably appear swollen in this state. The collapsed state (C) denotes an increase in specific surface of the organelle. In this case, the tubuli and cisternae of the smooth or rough endoplasmic reticulum are narrowed, whereas in mitochondria we observe shorter distances between the cristae.
3. Conditions A condition is a statement which is either true or false. Direct measurements and derived conclusions must be expressed in the form of conditions before they are suitable for handling in our system. Pathologic and morphologic states for a given organelle are designated in organelle mnemonic
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(N, S, R, M, C, P, L, or H) first, followed by the pathologic state mnemonic (N, P, T, R, 0, A, E, D, Y) or morphologic state mnemonic (G, S, C, U), respectively. For example MT denotes the mitochondria are hypertrophied and RS denotes the rough endoplasmic reticulum is vesiculated. Measurements require an additional symbol denoting quantity: J = significantly decreased, - = normal, 1 = significantly elevated. For example, SVT denotes “the smooth endoplasmic reticulum volume per unit volume cytoplasm is significantly elevated”. The negation of a condition is given by
TABLE 2
Mnemonic letters for quantitative organelle pathology Z = V = F = Q= G = S= C = U = W = 1= 2= 4=
numerical density volume density surface density specific membrane surface unchanged morphology vesiculated collapsed vacuolated liver weight oligoploid normoploid polyploid
N P R T 0 A E D Y
= = = = = = = = =
normal status proliferation hyperplasia hypertrophy hyperplasia atrophy ageneration dysplasia dystrophy
prefacing the condition with a minus sign, -. Thus, -MT denotes the mitochondria are not hypertrophied, -RS denotes the rough endoplasmic reticufum is not vesiculated, and -SVt denotes the smooth endoplasmic reticulum volume per unit volume cytoplasm is not significantly elevated. Each condition is designated by a unique Quine number (as opposed to numbers which designate amounts). Table 3 shows the Quine number for each condition in our system. For example, MT = 145, RS = 052, and svt = 013. 4. Implications
All statements which are processed by our system are initially formulated as implications. Each implication is either absolute or relative. An absolute implication corresponds to a single condition which is always true. For example, the statement “it is always true that A” would be expressed as ==-A. A relative implication is a string of two or more conditions divided by *.
0
042
005
0
0
030 031 032
029
012 015 016 021 022
011
006
0
0
033 034
0
023 024
018
017
0
0
Q
0
0
055
0
051
047
0
0
0
0
056
0
052
048
0
0
0
0
057
0
053
049
0
0
0
0
058
0
054
050
0
0
Morphology GSCUNPRTOAEDY
118
117
116
115
114
113
112
111
128
127
126
125
124
123
122
121
138
137
136
135
134
133
132
131
148
147
146
145
144
143
state
158
157
156
155
154
153
168
167
166
165
164
163
162
151 161
142 152
141
Pathological
Oligoploid = 043. Polyploid = 044. Liver Weight: f = 045, j, = 046. 0 = no measurements. In the morphometry section, the upper Qume number corresponds to f, the lower Quine number to 4.
040
039 041
037 038
0
020 027 028
Lysosomes = L
0
02.5 026
019
035 036
= M
0
013 014
001 003 004 009 008 010 002 007
Morphometry V F
Perosisomes = P
Cristae = C
Chondrioma
Rough ER = R
Smooth ER = S
Hepatocyte = H
Nucleus = N
Z
Quine numbers for each condition
TABLE 3
178
177
176
175
174
173
172
171
188
187
186
185
184
183
182
181
198
197
196
195
194
193
192
191
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For example, A 3 B which means, “whenever A is true, then B is true”. Relative implications can be compounded to desired complexity. For example, A*BvC A*B&C
(9 (ii)
A & B * (C & D) v (E & F)
(iii)
(A * B) * (C v D) & (E * F)
(iv)
The left side of the major * sign is called the hypothesis; the right side is called the conclusion. “Logical and” is denoted “W; “logical or” is denoted “v”. Eventually, these implications must be expressed in standard form. A relative implication is said to be expressed in standard form when the hypothesis contains exactly one condition, and all conditions in the conclusion are separated by logical ors. Statement (i) is already in standard form. Statement (ii) expands into two statements in standard form: A+-B and A * C. Statement (iii) becomes A*
-BvCvE A* -BvCvF A=> -BvDvE and A=+ -BvDvF. Statement (iv) becomes -A+CvD B*CvD --A* -EvF and B*
-EvF.
5. Nullities
In his Theory of Deduction, Quine (1948) presents the concept of nullity. A nullity is a statement of what does not exist (or is not true). Quine uses the symbol “y” to preface each nullity, but we shall dispense with it without loss
640
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of clarity in this paper. Qume states : “Let us write ‘yl? to mean ‘there are no F’...‘ - yF’ denies that there are no F, and thus affirms that there are F.” Similar devices for dealing specifically with biomedical data are discussed by Feinstein (1967) and Ledley (1965). The relative implication 1 =+ 2 means that conditions 1 and -2 cannot coexist. The nullity signifying “nonco-existence of 1 and -2” is expressed in shorthand as 1 -2. Similarly, the implication 1 * 2 v 3 is equivalent to the nullity 1 -2 -3. An absolute implication, say => 1, is equivalent to the nullity of opposite sign: - 1. In other words, “it is always true that 1” is equivalent to, the “nonexistence of - 1”. All implications written in standard form are converted to nullities in this fashion. Any collection of nullities is called a nullity set. The size of a nullity is the number of explicitly state (nonzero) conditions in the nullity. A nullity of size one is called a monoconditional. A nullity of size two is called a biconditional. A nullity of size k is called a k-conditional. There is also a nullity of size zero, which will be discussed further below. It is called the “o-nullity” or simply “w” because its presence in the system terminates all calculations and effectively spoils the sense of the system. Extensive effort may be required to establish the absence of o in a given system. In a system into which no data have been introduced, a monoconditional is undesirable, since it indicates the presence of a condition which is predetermined, without reference to data. This doesn’t formally spoil the system, but it is a warning suggestive of poor input logic. Biconditionals one of whose nonzero members is a label (experiment number) is a device both for introducing data and extracting conclusions relevant to specific experiments. 6. Null Addition
The fundamental operation which we perform upon pairs of nullities is called null-addition, denoted CD.The importance of this operation will be discussed below. First, the conditions in each nullity are lined up in ascending order of absolute value, with either zeroes or blanks representing the empty columns in the nullity. For example: 1 2 3 2
3 4
are properly lined up. Null addition is possible only if there is exactly one column in which one nullity has the positive element and the other nullity has the negative element (“sign reversal”). For example, null-addition is not possible on the above nullity pair, because no column exhibits sign reversal.
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On the other hand, this nullity pair 1 -2
3 2
-3
4
is unsuitable for null-addition because more than one column (second and third) have sign reversal. This nullity pair 1 -2 2
3 3 4
can be null-added. The formula for null addition is 0 a a a
@ o=o 68 a=a 8 -a = 0 o 0=0 63 a=a
(a # 0).
Thus, 1 -2 8
3
2
3 4 3 4
1
The nullity, 1 3 4, is called the null-sum. The special case in which the oaly nonzero column present in both nullities is the sign reversal column results in a null-sum of “w”. For example, -7 8
7
The significance of w is that its presence signals a contradiction within the system. We cannot be certain that w is absent from our system until we have performed exhaustive null additions, to obtain the canonical form. For economic reasons, we employ two shortcuts in our computer algorithm : we retain only (i) null-sums which contain an experiment label and (ii) nullsums which contain no more members than the larger-sized of the two nullities which are being summed. Both shortcuts have mathematical justification (Moore et al., manuscript in preparation). For label x (i.e., experiment number x), the former shortcut guarantees that all label-speczjk conclusions (but not necessarily all conclusions) will be found. A conclusion is said to be label-specific if it is true for label x, but not true for some conceivable experiment whose data are inconsistent with label x. All the conclusions which interest us in this model for quantitative organelle pathology (save for the o-nullity, which was ruled out separately) are T.B. 42
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label-specific. The second shortcut is valid when all label-specific nullities turn out to be biconditional or monoconditional. This requirement was satisfied by all our data sets. 7. Example Tables 4-6 present a worked out example of the nullity method applied to morphometric data. Experiment 928 involved rats with oncocytic transformation of the cytoplasm (Riede et a/., 1975a). Table 4 shows the actual TABLE 4 Example of morphometric analysis on rats with oncocytic transformation of the cytoplasm? (Experiment 928) Mnemonic
Symbols Quine Nos
NZT
001
NVJ NF+ HZ+ HV-
J-9 SVJ SFw
007 -009
-010 011 014
-015
LZJ LVff
040 -041
Wj,
-042 046 043
RQt MZt MV? MW CW
CQt PZJ.
Oligoploid
-
005
PVJ
RVJ RF’?
Control rats
0.23 0.049 6.2
0.58 0.026 6.3
m2/cm3 cm3/cm3 m2/cm8
31.7 0.07 3.2 35.2 320~10~ 0.22 2.06 4.7 15.7 145x log
91.2 0.02 6.7 125.5 525x lo9 0.53 15.2 29.1 23.5 65 x lo9 om9 20x109 oaO5
m2/cm3 cm3/cm3 m “/cm 3 m2/cm3 cme3 cm3/cm3 m2/cm3 ma/cm3 ma/cm3 cmm3 cm3/cm3 cme3 cm3/cm3
0.015
liver weight of diploid: tetraploid
Dimension
0.091 278~10~ 0.80
45 x 109 0.005
I:atio
Rats with oncocytic transfonnationf
cmF3 cm3/cm3 m2/cm3 cm-” cm3/cm3
162x lo6 0.052 0.037 160~10~ 0.85
004
-016 017 020 021 023 025 027 029 031 033 036 038
SQt
Morphometry
7.5 g 60:40
345x106
0444
3.og 95:5
nuclei
t Riede et al., 197%. $ Mean values from five animals each. S.E. < 10% of the mean. Symbols for the morphometric analysis described by Weibel et al., 1968; Weibel, 1969; Rohr & Riede, 1973.
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data for this experiment and the appropriate control values. For example, the volume, surface, and specific surface is given for the smooth endoplasmic reticulum. Volume is significantly decreased (control = O-049, experimental = O-026), which corresponds to the mnemonic condition SVJ and the Quine number 014 (Table 3). Specific surface is significantly elevated (control = 31.7, experimental = 91*2), which corresponds to the mnemonic condition SQt and the Quine number 017. Surface is essentially normal (control = 6.2, experimental = 6.3). Rather than construct a separate Quine number for SF-, we observe that SF- is the same as stating that - SFf and - SFJ (normal is equivalent to not-elevated and not-decreased). Thus experiment 928 gives rise to the following relative implications (by way of actual measurement):928 * 014and928 =P -015and928 * -016and928 =+ 017. In nullity form, we have 928 -014 and 928 015 and 928 016 and 928 -017. The complete set of input nullities for experiment 928 (i.e., based upon actual measurements) is summarized in Table 5(a).
TABLE
5
(a) Input nullities for experiment 928. (b) Output nullities for experiment 928 (a> 928 928 928 928 928 928 928 928 928 928 928 928 928
-001 -004 -005 -007 E -011
-014 015 016 -017 -020 -021
0.2
__928 928 928 928 928 928 928 928 928 928 928 928 928
-023 -025 -027 -029
-031 -033 -036 -038 -040 041
042 -046 -043
928 928 928 928 928 928 928 928 928 928 928
-049
-053 -057 -191
-162 -163 -194
-135 -136 -157 -178
The input nullities in Table 5(a) are entered into program CANON, which works out the canonical form as described above. The conclusions for experiment 928 are summarized in Table 5(b). For example, the nullity 928 -049 corresponds to the following relative implication: 928 =z. 049. In mnemonic notation, 928 => SC; in other words, “the smooth endoplasmic reticulum is collapsed in experiment 928”. The data and conclusions for experiment 928 are summarized in Table 6.
Nucleus= N Hepatocyte= H SmoothER = S RoughER = R Chrondrioma= M Cristae= C Peroxisomes = P Lysosomes = L Ploidy (1, 2, 4) L. weight= W
Morphometric Morphology
ZVFQGSCUNPRTOAEDY
Morphometry
data and conclusions for experiment 928. Description: cytoplasm. (Riede et. al., 1975)
TABLE 6
Pathological
state
oncocytic transformation
of
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8. Results Computer analysis shows that every definition in our system is nonvacuous (i.e., there exist some conditions under which the definition is satisfied). Within each organelle, every pair of definitions is mutually exclusive with complete data. We have analyzed data from liver parenchymal cells undergoing five different metabolic disturbances: inhibition of protein synthesis by cycloheximide (Riede, Sebass & Rohr, 1971), hypothyroidism (Riede et al., 1975b), exsiccosis (Riede, Kreutzer, Robausch, Kiefer & Sandritter, 1974), vitamin E deficiency (Riede et al., 1971), and oncocytic transformation in the cytoplasm (Riede et al., 1975a). Our analysis shows that it is possible to specify the pathologic states for all organelles and the hepatocyte as a whole, as well as the morphologic states for the tubulocisternal organelles (rough and smooth endoplasmic reticulum, mitochondrial cristae). For each of the seven organelles in each of the five experiments, there exists one and only one pathologic state; for each of the tubulocisternal organelles in each experiment, there exists one and only one morphologic state. 9. Discussion The introduction of definitions from general pathology into quantitative organelle pathology is important, but requires that we employ precise terminology. Our investigations with a model of organelle pathology suggest that it is possible to predict the morphological and pathological status of a cell largely on the basis of combinations of a few measurements. In other words, these morphometric measurements inherently contain virtually all the information we expect to gain from a simple description of cell and organelie status. Since the morphometric measurements are objective data and the computer logic is the same for every experiment, we have the added security that we are not prejudicing the interpretation of any particular experiment in terms of subjective expectations. Of course the computer logic itself is based upon certain expectations, but this logic is at least completely specified and internally consistent. The concept of proliferation @roles ferre-bear offspring) can be developed both for particulate organelles and for tubulocisternal organelles (endoplasmic reticulum and mitochondrial cristae). Proliferation at the organelle pathologic level is seen as a simple increase in organelle number or membrane surface (= numerical increase in membrane units), and often accompanies the early phase of sublethal cell damage (Riede et al., 1971; Rohr & Riede, 1973; Chayen & Bitensky, 1973). In organ pathology, hypertrophy refers to an increase in cell size along with changes which increase the size of the organ
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(Goss, 1966; Robbins, 1974; Walter & Israel, 1974). This concept is readily carried over into quantitative organelle pathology. The same holds for hyperplasia, which is defined in cellular pathology (Virchow, 1858) as an increase in the cell number in an organ or tissue with simultaneous volume increase (Goss, 1966; Richter, 1974; Robbins, 1974; Walter & Israel, 1974). The concept of atrophy is used to describe conditions under which a shrinkage of cell size follows loss of the cellular substance (Robbins, 1974; Walter & Israel, 1974). This also carries over to the realm of quantitative organelle pathology. In quantitative organelle pathology, we characterize “ageneration” as a numerical or surface reduction of an organelle without simultaneous volume changes. Organelle neogenesis stops, and we see either abnormally large organelles or abnormally dilated compartments (cf. Riede et aZ., 1971, 1975b). Concepts of dystrophy and dysplasia in organ pathology vary from author to author (Walter & Israel, 1974). In quantitative organelle pathology, we characterize these concepts as opposite changes (one increasing, the other decreasing) in the volume versus the number, or in the volume versus the membrane surface, of the organelle. We can characterize all possible changes in quantitative organelle pathology with these eight concepts in addition to the normal state. As shown by our computer analysis, the individual pathologic conditions are defined by logically nonvacuous implications. Evaluation of over 70 morphometric analyses on liver cells undergoing various metabolic disturbances and cell damage conditions (Riede & Moore, in preparation) has shown furthermore that our nine concepts for the pathologic state of the organelle have sufficiently sharp definitions that no overlap is possible between concepts. The assignment of morphometric data to various morphologic states is very time-consuming in the absence of a computer, and subject to many unconscious errors. Thus computerization disciplines our thought processes about organelle pathology, and provides us with a convenient tool for applying these insights to a large mass of data. Finally, the existence of a collection of logically consistent definitions and implications allows us to address deeper questions in organelle pathology: what reaction patterns are detectible over large bodies of experimental data? An approach to this question is the subject of our next paper. 10. The Mathematical
Argument
We can sum up the mathematical approach to our model by saying that we think about our problems with implications, but we “think about thinking” with truth tables. A truth table is a set of entries where each entry expresses a true-false status for each condition in the system. In a maximal truth table,
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every possible entry is present. For example, the maximal truth table for three conditions (numbered 1, 2, 3) would have eight entries, as follows: 1 2 2 1 1 -2 1 -2 -1 2 2 -1 -1 -2 -1 -2
3 -3 3 -3 3 -3 3 -3.
The maximal truth table for IZ conditions would have 2” entries. An implication, such as 1 =S 2, is equivalent to the nullity, 1 -2. In the truth table world, this means that all entries in the initial truth table which simultaneously containing 1 and -2 must be deleted. In the above truth table, this means deleting lines 1 -2
3
and 1 -2 If we also have the implication 2 3), then we must delete lines
-3.
3 =z=-2 in our system (equivalent nullity, 1
2
3
2
3.
and -1 By the rule of null-addition, follows :
we can combine nullities
I -2 and 2 3 as
1 -2 es 2 3 1 3’ This new nullity,
1 3, gives us the right to delete lines 1
2
3
and 1 -2
3
from the truth table. But of course we have already done so. In other words, null-addition merely generates tautologous nullities. However, these new
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nullities may provide us with insights we might otherwise not have had, as we shall see below. The reason we must resort to mathematics is that while the truth table is an indisputably complete description of any diagnostic system, it is also very cumbersome. Thus we shall develop computationally complete algorithms in the nullity world, then justify them in the truth table world. In studying a system containing n conditions, it is convenient to Quinenumber the conditions consecutively from 1 to n. Instead of the usual representation, we denote each nullity with an n-vector containing only Is, OS, and - 1s. Each positive condition, i, in a given nullity appears as a 1 in the ith column of the corresponding n-vector. Each negative condition, -i, in a given nullity appears as a - 1 in the ith column of the corresponding n-vector. All other colunms in the n-vector are zero. The zero-vector is known as w. Null addition proceeds as before. For example, in a seven-condition system, the null addition 1 -2 3 8 2 3 4 1 3 4 would be represented n-vectorially
as
(1, - 1, 1, 0, 0, 030) Q (0, 1, 1, LO, 090) (1, 0, 1, l,O, O,O) Entries in the truth table are similarly represented. For example, the truth table entry 1 -2 3 4 - 5 6 - 7 would be represented n-vectorially as (1, -1, 1, 1, -1, 1, - 1). We denote the maximal truth table (no entries deleted) as X. If r and s are two n-vectors, then we say that Y covers s, denoted Y < s, if for every ri # 0, 1 I i I n, si = ri. The relationship between a nullity set, S, and the truth table, T, which is represented by S, is that t E T if and only if there exists no s E S such that s covers t. We denote the statement S represents T by writing r(S) = T. A variety of different nullity sets may represent the same truth table, T. It is desirable to find a representation for T which is in some sense more compact than any other nullity set. For one thing, if distinct r, s E S and r I s, then s is clearly a redundant member of S. In other words, for any t E X-T such that s I t, r I t as well. Taking this notion a bit further, let us imagine the maximal set R such that r(R) = r(S). Now remove all redundant elements in R. This new set is called the canonical form of S,
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denoted /ISI/. Since this set IIS]I is unique, we say that ]Is// = r-r(T). The burden of our mathematical argument will be to demonstrate that IIS/ results from progressively expanding S by means of null-adding all possible pairs in S to exhaustion, with removal of redundant elements as they appear. The definition in our system are summarized below: Definition 1. (Nullity set). S is a nullity set if and only if for every s E S, s = (sl, . . . , s,) and xi = -1, 0, or 1, for every i, 1 I i < n. Definition 2. (Truth table). T is a truth table if and only if for every t E T, t = (t1, . . . , t,) and ti = -t 1 for every i, 1 I i I n. The maximal T is
denoted X. X is akin to the unit cube in graph theory (Deo, 1974). DeJinition 3. (Null addition). Let q, r E S- (0). Then s = q o YE S if and only if, (i) there exists a unique j, 1 I j I n, such that qj = -rj # 0 and sj = 0, (ii) for every i, 1 I i # j I n, such that qi # 0, si = qi, and (iii) for every i, 1 I i # j I n, such that ri # 0, si = ri.
4. (Cover). Let r, s E S. Then r I s if and only if for every = ri.
Definition
Yi # 0,
Si
Definition 5. (Representation).
if and only ifs I Definition
Let s E S. Then r(s) = T c X when t E X- T
t.
For nullity set S,
6. (Representation). r(s)
=
n S&S
r(s)
Dejnition 7. (Redundant element). For nullity redundant element in S if r I s and r # s.
set S with r, s E S, s is a
Definition 8. (Canonical form). For nullity set S, the canonical form for S, denoted IIsII, is that set such that if R is the maximal set for which r(R) = r(S) and Q is the set of redundant elements in R, IIsll = R- Q.
Our theorem shows that the canonical form for S is obtained of successive null-additions and redundant element removals.
by
a process
Theorem. (Canonical form). For nullity set S, let r(S) = T and R be the maximal set such that r(R) = T. (i) Let P be obtained from S by any sequence of null-additions and redundancy removals. Then r(P) = T. (ii) Let r E R. Then for each p 2 r there exists a P obtained from S by null-additions and redundancy removals such that p E P.
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Proof. Part I. Without loss of generality, let P be the first set obtained from S by a sequence of null-additions and redundancy removals such that r(P) # T, and let Q be the immediately preceding set [where r(Q) = T]. Obviously a redundancy removal could not change r(Q). Consider the q, r E Q such that r(Q u {q @r>) # T. There must exist a t E T such that s = q Q r I t. By Definition 3(i) of null addition and without loss of generality, there exists a uniquej, 1 5 j I ~1,such that qj = 1 and rj = - 1. Without loss of generality, let tj = 1. Since q $ t, by Definition 4 of cover, there exists an i, I I i # j < iz, such that 0 # qi = - ti. By Definition 3(ii) of null-addition, si = qi = - ti. By Definition 4 of cover, s $ t. Contradiction. Part II. Without loss of generality, order the n conditions such that all conditions i at which ri = 0 precede all conditions j at which rj # 0. Consider the first h at which there exists t 2 Y, u 2 r such that t, = 2kh# 0, ti = Ui = 0 for i < h, and ti = Ui # 0 for i > h and no p E P obtained after exhaustive null-additions such that p < t and p I U. Consider the distinct q, s E P such that q I t and s I u. Then qi = Si = 0 for i < h, qh = t, = -u, = -s, (or else a p E P would exist), and qi = ti if qi # 0 and si = ti if si # 0 for i > h. By Definition 3 of null-addition, p = q Q s exists and satisfies the above properties. Contradiction. This work was supported in part by a National ScienceFoundation grant to ProfessorM. Goodman and in part by the DeutscheForschungsgemeinschaft. We wish to thank Miss Elaine Krobock and Miss Karin Bensing for secretarial assistance. REFERENCES BOLENDER, R. (1974).Fedn. Proc. Fed. Am. SOCK exp. Biol. 33,2187. CHAYEN, J. & BITENSKY, L. (1973).In Cell BioIogy in Medicine (E.
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