Karl Weissenberg - The 80th Birthday
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The Weissenberg Rheogoniometer Adapted for Biorheological Studies
PROFESSOR A. L. COPLEY and R. G. KING
Laboratory of Biorheology, Departments of Medicine and
Pharmacology, New York Medical College, New York,
N.Y. 10029, U.S.A.
In 1957, one of us (A.L.C.), while working in London, approached Professor Karl Weissenberg about the possible use of the Rheogoniometer, named after him, for hemorheological studies. However, the apparatus, commercially available at that time (Farol Research Engineers Ltd. and, later, by its successor Sangamo Weston Controls Ltd., Bognor, Sussex, England), had to be modified for biological research. Although these modifications were made for hemorheological studies, they can be used as well for biological materials other than blood and its cellular and plasmatic components.
1. MODIFICATIONS OF THE RHEOGONIOMETER
In applications of the Weissenberg Rheogoniometer for hemorheological and other biorheological studies considerable modifications had to be made 1,3. They consisted of (a) the manufacture of a Mooney type geometry in Plexiglass, so that the sample can be observed throughout the test period; (b) special torsion bars were made to increase the sensitivity of the torque measurement; (c) the construction of a fluid damping device to reduce the noise/signal ratio of the torque measurement; (d) the application of a special spring to the normal force measuring system to increase its sensitivity; (e) the development of a special double Couette geometry, so that the torque derived from surface layers of plasma proteins can be measured; (f) an accessory, developed in conjunction with Dr. N. Siskovic and others of the Bioengineering Group at the Newark College of Engineering, consisting of a microscope with a television camera and monitor4. This attachment to the Rheogoniometer allows the microscopic observation of the behaviour of the cellular components of whole blood, such as red blood cells, when undergoing shearing on the viscometer cone.
2. TECHNIQUES AND CONSIDERATIONS FOR HANDLING BLOOD SYSTEMS
The use of blood or systems of its components in a viscometer requires the observance of several precautions:
(1)Blood has a corrosive action on metals and some other materials. Metal, conventionally used for those parts of the viscometer, which are in contact with the test fluid, could not be applied.
(2) Blood cellular elements tend to form clumps or clots and the plasma coagulates rapidly after the blood is shed. These processes can be slowed by siliconization of those parts of the viscometer which come into direct contact with the test material and by maintaining, in the case of whole blood, the sample at a temperature of about 4oC. Certain anticoagulants, however, will have to be employed to prevent the coagulation processes of fibrin formation and gelation in the fluid phase of blood, viz, plasma, as they do not necessarily prevent blood cellular clumping or aggregation. The action of the shear in the viscometer cup, particularly if prolonged, can cause the formation of macroscopically visible small clots, which would not be seen unless the viscometer geometry was transparent. These clots, which were microscopically found to be usually composed of platelets, cause the measurements to be unreliable. it is, therefore, important that the test sample be observed at least macroscopically during the measurement.
(3) Blood, systems of blood constituents, as well as other biological fluids necessitate the smallest possible volume, compatible with instrumental requirements for accuracy of measurement.
(4) Biological materials, which quickly degrade, must be measured in the shortest possible time.
(5) Some fluids, because of their small available volume from one subject, have to be obtained from several subjects and pooled, provided there is no biological objection to such a procedure. The smallest amount of fluid, that can be efficiently tested in the Weissenberg Reogoniometer, is 0.1 ml. However, the workable shear rate range, in testing such a small volume, would depend on the consistency of the fluid.
3. INSTRUMENT TECHNIQUES AND ACCESSORIES
Fig. 1 is a diagrammatic illustration of the combined Couette and cone and plate geometry, used when measuring the flow properties of whole blood, plasma and plasma protein solutions. We employed three sizes of this geometry. One size requires 45 ml of sample and two smaller versions, requiring 15 and 5 ml, respectively.
Fig. 1. Diagram of geometry used showing position of removable guard-ring.
The geometry is manufactured in Plexiglass and polished to be transparent so that the test sample can be macroscopically observed throughout the test period. Thermistors measure the temperature of the inner cylinder. An isothermal jacket, containing circulating water, surrounds the test platens so that varying temperatures can be used in securing experimental data.
A special, thin torsion bar made of beryllium copper alloy is used to give increased sensitivity. A dampening device is employed to reduce the “mechanical noise” or vibrations, which affect the measuring transducer. This dampening device consists of a small tank, containing oil, into which a paddle, attached to the torque measuring system, is dipped. Varying degrees of dampening can be achieved by employing oils of different viscosities in the tank.
The maximum sensitivity of the standard Weissenberg Rheogoniometer, as secured from the manufacturer, was increased from 0.01 to 0.001 dynes/cm2 in using our above described modifications. This permitted viscosity measurements on whole blood at minimal shear rates down to 0.0009 sec-1 and, occasionally, down to 0.0006 sec-1 (5_11).
4. MEASUREMENTS OF WHOLE BLOOD
Considerable discussion exists in the literature with regard to the significance of viscosity measurements at minimal shear rates, where the flow of blood approaches a stand-still and the shear rates progress to zero. Blood viscosity is markedly augmented in this situation and also in conditions such as inflammation, circulatory shock and during surgical operations.
For obtaining data on the flow properties of whole blood in the shear rate range from 1.0 x 10-3 to 1.0 x 103 sec-1, blood was drawn from healthy human donors. The blood was anticoagulated with EDTA (1.2 mg/mI), sodium heparin (10 NIH units/ml) and Paul’s oxalate mixture (1.2 mg ammonium oxalate plus 0.8 mg potassium oxalate per ml) used in dry form to exclude dilution of the test sample. We found no significant change in viscosity values obtained when using the three different anticoagulants in rheogoniometric measurements.
Fig. 2 is a typical flow curve of a blood sample of 44 per cent hematocrit (cell volume). The curve is representative of the data secured from eighty healthy blood donors, varying in age from 25 to 60 years.
Fig. 2. Typical flow curve of whole human blood divided into three regions together with examples of recorder traces.
We divided the shear rate scale into three regions in accordance with the rheological properties observed: Region I from 50 to 1000 sec-1, Region II from 0.01 to 50 sec-1 and Region III from 0.001 to 0.01 sec-1.
In Region I (50 to 1000 sec-1) whole blood exhibits Newtonian behaviour or nearly so. It is assumed that the aggregating force, which causes the formation of rouleaux, is smaller than the des-aggregating shear force. The latter causes the break-up of rouleaux, which have formed, into individual red blood cells. A typical example of a recorder trace, measured in this region, is shown in Fig. 2.
In Region II (0.01 to 50 sec-1) the blood viscosity becomes dependent on the rate of shear and also the time of shearing, with the blood showing thixotropic behaviour. After the rotational motion has been stopped, the time-torque curve does not return to zero, indicating that the blood has a yield stress. A typical recorder trace in this region is also shown in Fig. 2.
In Region III (from 0.001 to 0.01 sec-1) the viscosity curve approaches a plateau. The recorder trace of a measurement in this shear rate range shows no dependency on time of shearing as in Region II. We postulate that the shear stress in this region are below the yield stress and that a structure of rouleaux of red blood cells exists. The blood flows as a solid plug which may slip at the boundary of the blood viscometer wall interface on a plasma layer. From Fig. 2, at a shear rate of about 0.01 sec-1, a change in the slope of the curve can be seen. We consider this change of slope to demonstrate the point, where the plug of red blood cells breaks up and the shear stress measured represents real yield value of the blood.
(Footnote) In his publications, Karl Weissenberg related normal stresses to elastically recoverable shear strain. According to his theory, one should expect an undetectably small normal stress. He also stated that as far as he knew, this elastically recoverable shear strain is so small that it is not possible to detect it. This section of the present contribution gives support to Weissenberg’s theory, because according to the Reiner-Rivlin theory the normal stresses are related to the non-linear shear viscosity, and therefore one would expect a strong normal stress produced by the strong non-linearity of the viscosity of whole blood. This footnote is based on information given by Professor Karl Weissenberg in his letter of 12 June, 1973, to A. L. Copley.
5. MEASUREMENT OF SURFACE LAYERS OF PLASMA PROTEINS
Surface layers form in systems of solutions of fibrinogen and other plasma proteins, and also in the plasma itself 10-18. The measurement of the rheological properties of these surface layers has largely been avoided by most investigators with the exception of Joly19, because they considered them to be artefacts20-22.
Fig. 3 shows a plot of viscosity measurements of human plasma in a Couette type geometry with the detachable guard-ring in Fig. 1. We found Newtonion characteristics with the guard-ring in place. However, when the guard-ring is removed and the experiment is repeated, a non-Newtonian flow curve is obtained, with marked elevation of the torque (t) at the lower shear rates. From these two curves we can deduce that a layer of plasma proteins forms at the sample-air interface.
Fig. 4 is a plot of results obtained when a solution of 0.04 and 0.4 per cent fibrinogen in 0.9 per cent NaCI solutions are measured in the same way. Non-Newtonian characteristics and markedly increased t values are secured without the guard-ring in place, indicating that the fibrinogen solutions produced surface layers of considerable strength at the sample-air interface.
For direct measurements of the torque, derived from these surface layers12 and to avoid the cumbersome double measurement described above, a special Couette type geometry was devised.
Fig. 3. Platelet-rich human plasma measured with and without a guard-ring.
This, shown in Fig. 5, consists of a ring, 9 cm in diameter, which just penetrates the surface layers of the test sample which is held in an annular groove. These surface layers were shown by us3 to exhibit an elastic modulus. The preferable experimental technique is therefore the use of the oscillatory mode of the instrument, but up to the present time we made only exploratory rotation testing with this new geometry.
Figure 6 shows an example of the data obtained with the new accessory (Fig. 5). Torque measurements were made of surface layers, formed with varying concentrations of solutions of human ץ-globulin in 0.9 per cent NaCI, together with added 0.4 per cent human fibrinogen. We found that ץ -globulin preparations of 96 to 99 per cent purity reduce markedly the t values of surface layers formed in systems of solutions containing fibrinogen18. It can be seen from this Figure, that, upon the addition of a 0.75 per cent concentration of highly purified ץ -globulin to a 0.4 per cent fibrinogen solution, a considerable reduction in torque is found. Amounts of 0.04 and 0.1 per cent ץ -globulin have only a minor effect in decreasing the strength of the surface layer, with the lowest concentration exhibiting the least reduction.
Fig. 4. Solutions of 0.4 and 0.04 per cent fibrinogen in saline measured with and without the guard-ring in place.
6. MEASUREMENT OF NORMAL STRESS IN WHOLE HUMAN BLOOD
It has been demonstrated that whole blood has a structure, consisting of plasma proteins and blood cellular elements (red blood cells, white blood cells and platelets) which lead to its non-Newtonian flow characteristics. One may, therefore, expect a normal stress effect to be present in a material which has such a structure. Accordingly, a very fine leaf spring was constructed for the detection and measurement of the normal force which is an exponent of an elastic fluid. The measuring upper platen was suspended from this spring and therefore normal stress generated on the plates will be indicated. Upon calibration by weights, it was found that the minimum sensitivity of the measuring system was 1 dyne/cm2, when using a plate, 10 cm in diameter.
A second method required the use of an upper plate together with a lower cone of 1o angle, both 10 cm in diameter. Small capillary tubes were inserted across the centre of the flat plate for the demonstration of the distribution of normal stress across the cone. The blood of six healthy human donors was measured in rotational testing.
Contrary to our expectations, no normal stress above 1 dyne/cm2 could be detected when using either of the above methods.
Fig. 5. Special Couette type platens for the measurement of torque (t) from surface layers of protein solutions.
7. RIGIDITY MODULI OF BOVINE FIBRIN GELS INITIATED BY THROMBIN
Studies were made by us of the rigidity modulus measured in unidirectional shear23 of thrombin induced fibrin gels. These gels were found to be elastic according to Hook’s law, when subjected to harmonic shear strains of very small amplitudes.
Fig. 6. A plot of torque (t) versus the rate of shear of ץ globulin solutions plus 0.4 per cent fibrinogen solution compared with a 0.4 per cent fibrinogen solution as control.
The tests were made by placing 0.05 ml purified thrombin solution, containing 4 N.I.H. units on a 1o cone of 5 cm diameter. Then, with manual fibrinogen of 95 per cent clottability in 0.9 per cent NaCl at pH 7.0, was added. The plate of the rheogoniometer was immediately lowered into position and the gel was permitted to set for a predetermined time, which varied from 1 to 150 min. The sample was covered with paraffin oil, white U.S.P., in order to prevent evaporation. All tests were made at 22oC. The cone was rotated at 0.012 radians/sec for 5 sec after the predetermined time elapsed and the resultant tangential torque was measured on a continuous strip chart recorder. The torque reading rose immediately to a peak and fell back, indicating the collapse of the fibrin gel. The rigidity modulus was calculated by employing the value of the peak.
Figure 7 is a semilog plot of the rigidity modulus (dynes/cm2 x 103) versus time in minutes. Each time point in Fig. 7 represents the average of five determinations made of separate samples of fibrin gels. This graph indicates that, within the first 60 min. of the conversion of fibrinogen to fibrin, we are dealing with a dynamic system in which two significant changes in the slope are observed. The changes in the slopes in Fig. 7 may be interpreted to be due to the release of the fibrinopeptides A and B, the formation of the fibrin fibres, the continuous formation of the gel and/or any combination of these processes. Other observations concern the rigidity modulus of plasma gels24.
Fig. 7. A plot of rigidity modulus versus time of fibrin gels initiated by the venom of the Brazilian snake Bothrops atrox.
It has been established that fibrinopeptides A and B are released during the conversion of fibrinogen to fibrin upon the action of thrombin 25,26. It was shown 26,27 that the release of fibrinopeptide A occurs prior to that of fibrinopeptide B. The latter release was found by him to be four times slower than the former. We suggested, on the basis of this study, that one of the effects of the release of fibrinopeptide B is the formation or opening of sites which are utilized to form a firmer gel. Thus, as a first approximation, the first slope may indicate the release of the A peptide and the second slope that of the B peptide. Alternatively, another explanation for the increase in the rigidity modulus could be related to the contraction of the fibrin polymers which would result in tightening the gel structure.
8. THE EFFECT OF TEMPERATURE ON THE RIGIDITY OF FIBRIN GELS INITIATED BY BOTHROPS ATROX
The effect of the crude venom of the Brazilian snake Bothrops atrox, in gelating a solution of bovine prothrombin-free fibrinogen was examined by measuring the changes in the rigidity of the gel with time28. The B. atrox venom was diluted with 0.9 per cent NaCl by 1:1000. Purified fibrinogen was prepared from Armour bovine plasma fraction I by the method of Blombiick and Blombiick29.
A 1o cone and flat plate of 5 cm diameter was used for these measurements. After 0.05 ml of the venom was placed on the cone, 0.65 ml of fibrinogen was added with manual stirring. The samples were allowed to polymerize for periods of 1 to 60 minutes. A unidirectional shearing motion was applied to the gel at a rate of 0.012 radians per second. The resultant tangential toique was recorded on a strip chart recorder. The torque reading rose to a peak in approximately 5 sec. and fell back, indicating the collapse of the fibrin gel. The peak value was used for the calculation of the rigidity modulus.
The data indicated that the rigidity of the gel is controlled by temperature, because it influences the rate of reaction of thromboserpentin, a term introduced by Copley, Devi and Banerjee30 for the enzyme present in the venom. From 10 to 24oC the rigidity of the gels remained constant. At temperatures above 25oC, a gradual decrease in rigidity was observed until at 45 to 50oC no rigidity could be detected. The decrease in rigidity with increasing temperature may well be caused by an increase in the activity of fibrinolysin, contained in the venom. This results in accelerated degradation and lysis of the fibrin.
9. MICROSCOPIC OBSERVATION OF FLOWING WHOLE HUMAN BLOOD AT VARYING SHEAR RATES
It has been demonstrated by Schmid-Schonbein, Wells and Goldstone31 that the rouleaux of red blood cells present in unsheared whole blood will desaggregate, when subjected to high shear rates. It would be desirable to follow the action of varying shear rates on rouleaux of red blood cells over a wider range of shear rates than was available to these investigators by the use of the Wells-Brook-field viscometer. A microscope with an associated television camera and monitor was therefore attached to the Weissenberg Rheogoniometer. Fig. 8 is a diagrammatic drawing of this device.
Preliminary microscopic observations were made with the accessory on the deformability of red blood cells and of their rouleaux under varying rates of shear from 10-3 to 103 sec-1. Microscopic studies on the behaviour of other cells such as, for instance, spermatozoa could much lower than hitherto possible with existing apparatus employing the Wells-Brookfield viscometer31.
Fig. 8. A schematic diagram of the arrangement of a microscope and T.V. camera when attached to the Weissenberg Rheogoniometer.
10. IN CONCLUSION
Our adaptations of the Weissenberg Rheohoniometer for the measurement of blood systems are described and some of our findings are reported. The adaptations can be applied also to other biological systems. The instrument, with the modifications which we made, has been found to be an excellent tool for studies of flow properties of blood systems and of other biological fluids.
As the use of the Weissenberg Rheogoniometer already proved to advance knowledge on blood, it will undoubtedly be useful in the study not only of blood, but of other biological fluids in health and disease. The application of the Weissenberg Rheogoniometer in the study of biological fluids can thus be expected to be of benefit to the health of the human species.
It is our privilege and pleasure to be among those who honour the numerous achievements and high attainments of Karl Weissenberg on the occasion of his eithtieth birthday. The many discussions which we had with him over the last sixteen years in England, France, Holland and at his repeated visits to A. L. Copley’s laboratories in the United States have been most helpful. They provided great stimulus to us and others associated with these laboratories.
A. L. Copley considers himself particularly fortunate to enjoy Karl Weissenberg’s friendship. He is also deeply grateful for Karl Weissenberg’s keen interest in his biorheological studies made the past thirty-five years including the past six years in which the Weissenberg Rheogoniometer has become a principal tool. We are wishing Karl Weissenberg many years of continuous productive scientific work in good health.
The research presented in this contribution were aided in part by the Contracts NR 2754 and N 00014-67-A-0449 of the Office of Naval Research and by the Department of Medicine and Surgery, Veterans Administration, Washington, D.C.
1. R. G. King and A. L. Copley, “Some Modifications of the Weissenberg Rheogoniometer for Adaptation to Hemorheological Studies.” In: Theoretical and Clinical Hemorheology, Proc. 2. Internat. Conf. University of Heidelberg, Eds. H. H. Hartert and A. L. Copley. Berlin-Heidelberg-New York, Springer-Verlag, 1971, p. 368-387.
2. R. G. King and A. L. Copley, “Modifications to the Weissenberg Rheogonlometer for Hemorheological and Other Biorheological Studies.” Biorheology 7, 1, 1970.
3. R. G. King and A. L. Copley, “An Accessory to the Weissenberg Rheogoniometer for the Measurement of Viscoelasticity of Surface Layers of Proteins.” Biorheology, 9:1972; Abstracts, I. International Congress of Bioerheology, Lyon 1972, p. 39; Biorheology 10: 1973 (in press).
4. N. Siskovic, R. Sinusas, J. L. Martin, R. G. King and A. L. Copley, “The Attachment of a Microscope and T.V. Camera to the Weissenberg Rheogoniomerer.” To be published.
5. A. L. Copley, “Some Problems in Hemorheology.” (Invited Lecture). Proc. 5. Internat. Congress on Rheology, Kyoto University, Ed. S. Onogi, Tokyo University Press, Tokyo, and University Park Press, Baltimore, Maryland and Manchester, England, 1970, Vol. 2, p. 3.
6. A. L. Copley and R. G. King, “Rheogoniometric Viscosity Measurements of Whole Human Blood at Minimal Shear Rates Down to 0.0009 sec-1.” Experientia 26: 904, 1970.
7. A. L. Copley, C. R. Huang and R. G. King, “Rheogoniometric Studies of Whole Human Blood at Shear Rates from 1000 to 0.0009 sec-1. Part I - Experimental Findings.” Biorheology 10, 17, 1973.
8. C. R. Huang, R. G. King and A. L. Copley, “Rheogoniometric Studies of Whole Human Blood at Shear Rates Down to 0.0009 sec-1. Part II. Mathematical Interpretation.” Biorheology, 10: 23, 1973.
9. A. L. Copley, C. R. Huang and R. G. King, “Flow Properties of Whole Human Blood at Minimal Shear Rates Down to 0.0009 sec-1.” Proceedings of the IV. Internat. Biophysics Congress, Moscow, U.S.S.R., 7-14 August 1972, in press.
10. A. L. Copley, “On Biorheology”. Biorheology, 9: 141, 1972; Abstract, I. Internat. Congress of Biorheology, Lyon 1972, p. 9.
11. A. L. Copley, “Hemorheological Aspects of the Endothelium-Plasma Interface”. Abstracts, VIII. Congr. European Soc. Microcirculation, Aberdeen, Scotland, Aug. 26-Sept. 1, 1972, P. 181.
12. A. L. Copley, “Non-Newtonian Behaviour of Surface Layers of Human Plasma Protein Systems and a New Concept of the Initiation of Thrombosis.” Biorheology 8: 79, 1971.
13. A. L. Copley, “Rheogoniometric Measurements of Overall Viscous Resistance of Plasma-Fibrinogen Systems and a New Concept of the Initiation of the Thrombosis.” Abstracts, 2. Congress Internat. Society of Thrombosis and Haemostasis, 11-17. July, 1971, Oslo, Norway, p. 70.
14. A. L. Copley, “Rheogoniometrically Measured Viscosity Profiles of Plasma Protein Systems and a New Concept of the Genesis of Thrombosis.” Abstracts, 25 International congress on physiological Sciences, July 25-31, 1971, Munich, West Germany.
15. A. L. Copley and R. G. King, 93Rheogoniometric Study of Viscosity of Fibrinogen and Plasma-Fibrinogen Systems.94 Federation Proc., 30: 480,1971.
16. A. L. Copley, R. G. King, “Viscous Resistance of Thromboid (Thrombus-Like) Surface Layers in Systems of Plasma Proteins Including Fibrinogen.” Thrombosis Research 1:1, 1972.
17. A. L. Copley and R. G. King, “The Action of Human Red Blood Cells and Platelets on Viscous Resistance of Plasma Protein Systems.” Biorheology 9:147, 1972; Abstracts, I. International Congress of Biorheology, Lyon 1972, p. 25; Biorheology 10: 1973, in press.
18. A. L. Copley and R. G. King, “Viscous Resistance Lowering Action of Highly Purified Gamma Globulin on Surface Layers of Fibrinogen and Other Plasma Proteins”. Abstracts, IV. Congress of the International Society on Thrombosis and Haemostasis. Vienna, 19-22 June, 1973.
19. M. Joly, “Etude par viscométrie superficielle de la déformabilité des macromolecules.” Proc. 5. Internat. Congress on Rheology, Kyoto, Japan, 1968. Ed. S. Onogi, Tokyo, University of Tokyo Press; Baltimore, Md. and Manchester, England, University Book Press. Vol. 2., p. 191, 1970.
20. R. E. Wells and E. W. Merrill, “Shear Rate Dependence of the Viscosity of Whole Blood and Plasma.” Science 133: 763 1961.
21. Chien, S. Usami, H. M. Taylor, J. L. Lundberg and M. I. Gregersen, “Effects of Hematocrit and Plasma Proteins on Human Blood Rheology at Low Shear Rates.” J. Appl. Physiol. 21: 81, 1966.
22. D. E. Brooks, J. W. Goodwin and G. V. F. Seaman, “Interactions among Erythrocytes under Shear.” J. AppI. Physiol. 28: 172, 1970.
23. A. L. Copley, R. G. King and B. M. Scheinthal, “Rigidity-Moduli of Bovine Fibrin Gels Initiated by Thrombin.” Biorheology 7: 81, 1970; 197, 1971.
24. A. L. Copley, A. Devi, R. G. King, B. M. Scheinthal and P. Ohlmeyer, “Gelation of Fibrinogen and Plasma Systems Studied by Light Scattering and Rheogoniometric Methods.” In: Theoretical and Clinical Hemorhology. Proc. 2. Internat. Conf. University of Heidelberg. Eds. H. H. Hartert and A. L. Copley. Berlin-Heidelberg, New York, Springer-Verlag, 1971, p. 154.
25. A. L. Copley, B. W. Luchini and E. W. Whelan, “On the Role of Fibrinogen-Fibrin Complexes in Flow Properties and Suspension-Stability of Blood Systems”. In: Hemorheology. Proc. I. Internat. Conf. University of Iceland, 1966, Ed. A. L. Copley. Oxford-New York, Pergamon Press, 1968, p. 375.
26. F. R. Bettelheim and K. Bailey, “The Products of the Action of Thrombin on Fibrinogen.” Biochim. Biophys. Acta 9: 578, 1952.
27. B. Blombäck, “Studies on the Action of Thrombic Enzymes on Bovine Fibrinogen as Measured by N-Terminal Analysis.” Arkiv Kemie 12: 33, 1958.
28. B. M. Scheinthal, R. G. King and A. L. Copley, “The Effect of Temperature on the Rigidity Modulus of Fibrin Gels Produced by Bothrops atrox Venom.” Biochim. Biophys. Acta 214: 260, 1970.
29. B. Blombäck and M. Blombäck, “Purification of Human and Bovine Fibrinogen.” Arkiv Kemie 10: 415, 1956.
30. A. L. Copley, S. Banerjee and A. Devi, “Studies of Snake Venoms on Blood Coagulation. I. The Thrombo-Serpentin (Thrombin-like) Enzyme in the Venoms.” Thrombosis Research 2: (No. 6), 1973, in press.
31. H. Schmid-Schönbein R. Wells and J. Goldstein, “Influence of Deformability of Human Red Cells upon Blood Viscosity.” Circulation Research 25: 131, 1969.
Preface / Acknowledgements / Biographical Notes
Weissenberg’s Influence on Crystallography
Karl Weissenberg and the Development of X-Ray Crystallography
The Isolation of, and the Initial Measurements of the Weissenberg Effect
The Role of Similitude in Continuum Mechanics
The Effect of Molecular Weight and Concentration of Polymers in Solutions on the Normal Stress Coefficient
Elasticity in Incompressible Liquids
The Physical Meaning of Weissenberg's Hypothesis with Regard to the Second Normal-Stress Difference
A Study of Weissenberg's Holistic Approach to Biorheology
The Weissenberg Rheogoniometer Adapted for Biorheological Studies
Dr. Karl Weissenberg, 1922-28
Weissenberg’s Contributions to Rheology
The Early Development of the Rheogoniometer
Some of Weissenberg's More Important Contributions to Rheology: An Appreciation
Publications of Karl Weissenberg and Collaborators / List of Contributors
© Copyright John Harris