Serge Gracovetsky, Ph.D

The Spinal Engine
This book has 271 pages. This document is a sample of the various chapters and appendices


This is a revised version (october 2008) of the original book I published in 1988. It has been reformatted in the standard 8.5x11 paper and is published in paperback with perfect cover. It contains 271 pages with 180 illustrations. It is catalogued under the ISBN 978-1-4276-2997-5.

This work describes the theory of the Spinal Engine and gives experimental evidence in support of the theory. I have encluded a 35 minutes movie in WMV and MOV formats describing how the function of the spine can be measured. The interested reader can watch:

1- A sample of the function of the collagen fibers of the annulus fibrosus;

2- And the role of the spine in locomotion.


Table of Contents







The Spinal Engine





Chapter One


The Evolutionary Record

The Critter Walk - Principles Of Motion

The Critter Walk - Mathematical Analysis

Analysis Of The Infinitely Flat Critter

Application To A Vertebrate Critter

The Critter With Short Stumps

Lateral Bending With Legs That Swing

Lateral Bending, Legs That Swing And Lordosis













Chapter Two



The Problem Of Spinal Modelling

Limitations Of Mathematical Modelling

What Is A Model?

Unavoidable Assumptions

Construction Of A Model - The Search For Hypotheses

Basic Types Of Mathematical Models Of The Spine

- The Continuum Models

- The Discrete Parameter Models

- The Muscular Response Models

The Musculoskeletal System

The Decision-Making Process

Application To A Realistic Spine Model


















Chapter Three



Functional Anatomy Of The Soft Tissues Of The Spine            


·                           Multifidus

·                           Longissimus Lumborum

·                           Longissimus Thoracis

·                           Iliocostalis Lumborum

·                           Iliocostalis Thoracis

·                           Internal/External Obliques

·                           Transversus Abdominis                                                

·                           Rectus Abdominis

·                           Latissimus Dorsi

·                           Psoas

·                           Quadratus Lumborum

The Posterior Ligamentous System

General View Of The Erectores And Spine Ligaments























Chapter Four

A Model Of The Spine In Flexion Extension: Mathematical Formulation

Construction Of The Model

·                           The Geometry Of The Spine

·                           The Lumbar Curve

·                           The Musculature

·                           The Posterior Ligamentous System

·                           The Lumbodorsal Fascia

·                           The Midline Ligament

Comparison Of The Roles Of The Fascia And The Midline Ligament In Sagittal Motion

Consequences Of Our Formulation On The Mechanics Of The Spine

Control Of The PLS: The Need For Lordosis

The Effect Of Time On The Efficiency Of The PLS

Determination Of The Contribution Of The PLS

Description Of The Control System

Computational Considerations

Detailed Description Of The Sub-Optimization Procedure

The Objective Function

Experimental Evidence Supporting The Concept Of Stress Equalization And Minimization


























Chapter Five



The Calculated Response Of The Spine In Flexion-Extension

Case One - Lifting While Lordosis Is Maintained

Case Two - Lifting While Lordosis Is Reduced

General Case - Lifting Using A Variable Lordosis   Throughout The Progression Of The Lift

Selecting The Best Lordosis

Freedom Of Movement

Holding Weights In The Erect Stance

The Teaching Of Lifting

The Muscle Relaxation Phenomenon

The Spine As A Megamuscle

The Centers Of Reaction And Rotation - Stress Equalization

Fundamental Assumption In The Evaluation Of The Centre Of Rotation

The Instantaneous Centre Of Rotation

The Centre Of Rotation

Relationship Between The Centre Of Rotation And The Centre Of Reaction

Control Of Lordosis

Predicted Pathology




























Chapter Six



The Measured Response Of The Spine In Flexion-Extension

The Intradiscal Pressure

The EMG Response

Intra-Abdominal Pressure (IAP)

Maximum Extension Moment Of The Erectores

The Range Of Trunk Motion

Experimental Procedure Chosen To Validate The Model

Clinical Evaluation

Experiment # 1: Relationship Between Lordosis, Angle Of Trunk Flexion And The Activity Of The Erectores

Experiment # 2: Sagittal Dead Lift

Experiment # 3: Impact Of Spine Injury On Spinal Function

Experiment # 4: The EMG Response

Experiment # 5: The Internal Abdominal Pressure
























Chapter Seven



The Response Of The IV Joint To Compression And Torsion


Mechanical Etiology Of Low Back Pain

Response Of The IV Joint To Compression

The Annulus Fibrosus


Response Of The IV Joint To Torsion

Modelling Of The Intervertebral Joint
















Chapter Eight



Coupled Motion: The Gearbox Of The Spinal Engine

Experimental Studies On The Coupled Motion

General Comments

The Role Of The Facets - A Gearbox

Solving For The Loci Of Left And Right Facet Contact

Remarks On The Solution Of The RFL And LFL

Coupled Motion - The Functional Context

Loss Of Disc Space And Facet Tropism
















Chapter Nine



The Theory Of The Spinal Engine


The Need For A Unified Theory Of Locomotion

Classical Theories Of Locomotion

Locomotion As The Product Of An Oscillating System Principle Of Work Of The Spinal Engine

The Smallest Functional Unit Of The Spinal Engine

Combining Two Basic Building Blocks

Combining Three Basic Building Blocks

Increasing The Efficiency Of The Engine

Increasing The Power Of The Engine

Exploiting The Earth’s Gravitational Field




















Chapter Ten



Power Flow In The Spinal Engine

General Approach To The Problem

Detailed Procedure

Modelling Of The Intervertebral Joint

Torque Required To Rotate An Intervertebral Joint In The Transverse Plane

Characterization Of The Axial Torque

The Time Constant Of The Disc

Effect Of Compression

The Average And The Individual

Comparison With Data Collected By Other Researchers

Torque Required To Rotate An Intervertebral Joint In The Transverse Plane During The Normal Walking Cycle

The Cappozzo Experiment

The Thurston And Harris Experiment

Distribution Of Torque Between The Annulus And Facets

General Comments

Implications Of The Working Hypothesis

Energy Transfers In The Frontal And Median Planes

General Predictions Of The Spinal Engine Theory



























Chapter Eleven



Test Of The Spinal Engine Theory – Locomotion

Comparison Of Locomotory Styles

Part # 1 – Kinematics

The Baseline For Comparison

Comparison Of Kinematic Data

Part # 2 – Comparison Of The Timing Of Muscle Activity

General Remarks















Chapter Twelve



Application Of The Theory

Application #1: The Determination Of Safe Loads

The NIOSH Biomechanical Standards

Determination Of A Safe Load Criterion

Application #2: The Baseball Pitcher

Application #3: A Recommended Exercise

Application #4: Some Surgical Practices















Appendix 1



The Three-Dimensional Model Of The Human Spine


The First Task - Anatomical Details

The Second Task - Spinal Posture Details

The Third Task - Mathematical Details

The Fourth Task - System Details

Characteristic Curve Motion Algorithm

Outline Of Optimization Algorithm














Appendix 2



The Spinoscope

Example Of Clinical Use

Overview Of The Problem Of Low Back Evaluation









Appendix 3




The Functional Anatomy Of The Intervertebral Joint

Components Of The Intervertebral Joint

The Intervertebral Disc

The Vertebral Body

The Facet Joints

Ligamentous Structures






























The Spinal Engine


“I found everything perfectly clear, and I really understood absolutely nothing. To understand is to change, to go beyond oneself. This reading did not change me.”


Jean-Paul Sartre


Search for a Method: Comment on Capital

and German Ideology by Karl Marx


Cybernetics! This is the age of the machine. Everywhere we look, living things are being described in terms of robotics. The bewildered physician is taken through differential equations, all kinds of Laplace and Fourier transforms, rational and irrational operators, and the like. At the end of the discussion, our perplexed man of medicine finds it difficult to contain his frustration and resentment towards such a mathematical cavalry.


Alas, in more cases than not, the problem solver loses sight of the initial problem. Like the sorcerer’s apprentice, his solutions sometimes develop an autonomy not originally intended. This unwanted by-product is often referred to as a “school of thought.”


The study of the human spine did not escape this predicament: one look at the literature shows it to be full of diehard concepts. Somehow an idea will survive any kind of experimental annihilation if it is attractive enough. When we ask ourselves why this is so, we find there are no easy and clear-cut answers.


The act of describing our environment is as old as mankind itself. We use our senses to perform measurements and, by so doing, take the first step in the process of understanding. The early Babylonians were famous for their ability to track the position of the stars. Mathematics became the tool of choice in their attempt to order the resulting data. Its usefulness was demonstrated by both Newton and Kepler, who reduced the vast number of available astronomical observations to a few simple relationships. This set of relationships is a model of celestial mechanics. Strict mathematical computations can be compared with the behavior of the real system and will ruthlessly expose any weakness inherent in the assumptions of the model. Conversely, such a model may also reveal any number of unforeseen mechanisms.


When the discrepancy between the measured orbit of Uranus, and its calculated orbit (using a model based upon Kepler’s laws), exceeded instrumentation error, astronomers were faced with two possible explanations: either the model (and its underlying assumptions) was inadequate, or another unknown planet was disturbing the orbit of Uranus. The French mathematician Urban Leverrier assumed the existence of another planet and, using the model, went on to calculate its position. His research led directly to the discovery of Neptune.


It should be noted that Leverrier chose to believe in the accuracy of the model, as it was consistent with the precision of the instruments of his time. Fewer than fifty years later, superior instrumentation led to the revelation that the model, itself a consequence of Newton’s laws, was still not in perfect agreement with measurement. The theory of relativity eventually led to the development of a better model.


There is no such thing as the “perfect” model. The best that can be achieved is the creation of a model that explains the data available at a particular point in time. Some models survive for a long time whereas others do not. Regardless, a model is no substitute for common sense. Rather, it is a tool to be used to express hypotheses and rigorously evaluate their consequences.


That is not to say that models are absolutely necessary. Galileo did not require a model in order to realize that the earth was rotating. To be a success, a surgical procedure need not be determined by the solution of some fantastic equation. Nevertheless, as humanity becomes more refined, the limitations and costs associated with a purely experimental approach become more pronounced. For instance, it is known that Neptune orbits the sun every 165 years, even though the planet has not been observed for that long, simply because the model speeds up the acquisition of knowledge.


This is the reason for modeling biological systems. With experimental methods not always providing data sufficient to solve the many problems in medicine and biology, the theorist can join forces with the experimentalist. The added dimension of the theoretical modeling approach can both lead to a further understanding of a given problem and also potentially provide solutions not otherwise attainable.


However, no one today would dare describe an entire animal in mathematical terms. On the other hand, there are some aspects of animal behavior that may be analyzed and simulated by means of tools produced from that branch of engineering known as “system theory”.


System theory is a collection of overlapping areas such as control theory, information theory, automation theory and communication theory. Although most of the basic concepts were formulated as early as the nineteenth century, it was the Second World War that provided the impetus necessary to express these concepts formally and apply them to the destruction of millions of human beings. Both automatic piloting for aircraft as well as accurate gunfire from warships lurching their way through rough seas necessitated control systems more sophisticated than Watt’s early steam regulator. In addition, the need to coordinate world-scale military operations propelled communication theory to new heights.


It was the race for the moon that triggered the massive research effort needed to ensure the proper guidance of spacecraft during orbit transfer. Flight conditions were almost ideal for the application of the precise laws of celestial mechanics, and so the control problem was readily expressible in mathematical form. Moreover, with the development of digital computers, it has been possible to model and calculate the trajectory of these space vehicles with extraordinary precision. These ideal mathematical conditions provided fertile ground for the frantic development of system theory. The first lunar landing went far to convince many of its worth.


The application of these new ideas to the medical profession was eased to a large extent by the parallels between engineering and biological control systems. Control theory was used to analyze the physiology of respiration, thermoregulation and ionic balance. Entire subsystems were considered in the description of behavior and development. For a while, success seemed to be just around the corner.


Not everyone shared this enthusiasm. Oatley (1973) commented that “the application of control theory to biological systems is seen by some as being quite essential to their understanding, something like the way the ability to count is necessary for knowing whether one’s change is correct. By others, it is seen as yet one more fad whereby those with some faintly exotic expertise can rephrase what is known already in terms that serve to obscure it”.


Such criticism is hard to swallow although some of it is, in fact, deserved. The orthopaedic surgeon confronted with a pain-wracked patient is hardly likely to accept the utility of a model invoking half of the Greek alphabet when no one will tell him what to do to improve his patient’s condition. It could be further argued that modeling, as practiced, might not be the best approach. One possible reason might be that mathematical models are greatly dependent upon fast digital computers, whereas the human brain is notoriously slow at computation. This alone should alert us to the realization that the mastery of Nature’s laws is not necessarily a matter of number crunching. This revelation is obvious whenever we attempt to model the reasoning process used to perform a diagnosis. The representation of knowledge is best achieved with the heuristic rules we intuitively follow.


The search for decision making strategies to order and process our knowledge has centered on the field of artificial intelligence. Yet this alone is not enough. The lack of adequate tools to integrate our knowledge leads to the existence of a wide range of possible mathematical models. This apparent infinity of designs runs contrary to observation as biological problems are solved by nature in very specific ways. The structure of the vertebrate spine must have been advantageous in evolution; its description by a wide variety of models is indicative of our failure to grasp the basic principles underlying the need for its very existence. After all, we cannot analyze the spine without considering its function.


Obvious as this may seem, mechanical tests are still conducted on the intervertebral joint on a daily basis, with little consideration being given to the function of the spine. Squeezing a joint in a press only indicates how the joint behaves in a press. This behavior may or may not be related to what the joint has evolved to do under normal physiological conditions.


What is the function of the spine? How can it be characterized in vivo? The magnitude of and ethical questions raised by a direct experimental analysis precludes any direct attack on the problem itself. Our experimental knowledge must be structured by some mathematical process. We propose to use system theory, or, more precisely, the theory of optimal control.


The Optimum Human Spine


The theory of evolution and natural selection implies the survival of the fittest and the elimination of the weak from the pool of genetic information available for the continuity of a species. What is the meaning of “fit for survival”, and what ranking system decides that the weak are, indeed, weaker than some required norm? Without entering into philosophical debate, it can be said that the ecological niche occupied by a species supplies the species with the energy it needs for its survival. In that case, the fittest can be defined as those members of a population that make the most economical use of the energy sources available in their ecological niche. It is our belief that energy efficiency is the absolute criterion for survival. This definition allows one to mathematically rank individuals of the same species.


One can even proceed a step further. If being the fittest is an expression of an individual’s overall performance, can one say that evolution has resulted in the creation of an optimum biological system comprised of optimally interconnected sub-systems?


From the point of view of survival, a biological system should be able to function, albeit at a reduced efficiency level, despite the loss of some of its parts or even of an entire sub-system. This feature is demonstrated by some lower forms of life, such as plants, which can be cut and mutilated quite extensively before the organism finally dies. It is reasonable to visualize the human body system as a collection of sub-systems, e.g. the musculoskeletal system, the cardiovascular system, etc., which are strongly interrelated in a way which still allows each sub-system to be characterized separately both by the task it has to perform and by the overall objective of the whole system. A particular sub-system is deemed to be optimum, or best, if it performs each particular task with the minimum consumption of energy.


Optimal performance of one or more individual sub-systems does not imply that the entire organism is also functioning at its optimum. There are strict mathematical conditions that must be satisfied to ensure this. A situation has attractive advantages when the local optimum of each sub-system occurs for the same parameter values that optimize the entire system when considered as one unit. Under these conditions, each sub-system can be  maintained close to its individual optimum without the benefit of complete global coordination. In a less drastic way, it would be desirable that a repair process, such as one for a minor internal bone fissure (a local event), be accomplished using mainly local resources.


There are other instances where this is not the case. The entry of the AIDS virus through a small incision is clearly not a local problem. In this particular case, a rigorous mathematical analysis of the ensuing global body response is beyond the reach of today’s technology.


However, there are other occasions where this type of analysis may well be possible. We would like to understand the conditions under which the normal function of the musculoskeletal system is governed by both the task to be accomplished and the need to do so with the minimum amount of energy.


So far, no one has expressed the function of the musculoskeletal system in such a generalized mathematical form. One must ask what simpler but parallel objectives the musculoskeletal sub-system might be accomplishing.


In the quest for such objectives, one can start with the hypothesis that survival means that the sub-systems will not self-annihilate. This implies that, at any given time, the level of mechanical stress within the musculoskeletal sub-system will not exceed some ultimate value. Regardless of the task being accomplished, the central nervous system will activate appropriate muscles to prevent such an event. Since the various components of the musculoskeletal sub-system have different mechanical characteristics, it is not unreasonable to speculate that the level of stress in all its components during the execution of a task will be proportional to the ultimate limit of each individual component. For example, if, for an arbitrary task, the stress within the bone reaches 2/3 of its ultimate, then the stress within the ligaments will also be 2/3 of its own ultimate.


This idea has a number of implications. As each vertebra is comprised of the same material, the stress within each should be equal. This is another way of saying that the task should be accomplished in such a way that the stress within the spine is equalized at its lowest level. This concept is not really new; the seeds of it were planted by Wolff as early as 1870.


Agreement on the specific task for which the spine has evolved must precede the development of this concept. Although there is no way of knowing for sure what that particular task might be, we propose that the most important activity for members of the vertebrate species is locomotion, either to get food, to avoid being eaten, or any of a number of other important reasons. It is appropriate to examine the stages of development of our ancestors’ locomotive ability. In so doing, the spine and its surrounding tissues emerge as the pervasive element - the primary engine - of locomotion in animals such as us.


It is obvious that the spine of a fish and its surrounding tissues represents the primary engine which the animal uses for locomotion. In analyzing the evolution of our fish-like ancestors, we seem to have lost touch with this fact. To this day, gait analysis is essentially the analysis of the motion of the legs. The legs are certainly useful, but are they essential? The answer is definitely no. Human bipedal gait will be demonstrated not to require the presence of any extremities. In retrospect, it was evident that the primary function of the spine, so obvious in the fish, was never transferred to any of our extremities during the long evolutionary journey.


Locomotion is but one of the many tasks that the human spinal engine is asked to perform. In many ways, tasks such as weightlifting require the use of the spinal engine for applications for which it has not been optimally designed. This study will examine the conditions under which the spinal engine can be safely used as a lifting engine.


From the theoretical and clinical analysis of the normal function of the spine, one can describe the changes which degrade its function and appreciate the reasons for observed modifications, such as facet arthropathy. This forces us to define normality in terms of functionality, rather than anatomical perfection. From there, a procedure for the objective measure of spinal function is proposed, and its use in the assessment of rehabilitation discussed. Indeed, the ultimate test of this theory rests with the ability to help the injured.


If history is any guide, the theory of the spinal engine will be vigorously challenged and eventually end up being replaced.


In the mean time, this theory is intended to replace a large number of local paradigms. It is hoped that this attempt at proposing a unified approach to the eternal back problem represents a step forward in the search for a solution.



Serge Gracovetsky, PhD.

Montréal, QC, Canada                                                         Original writing October 1988                                                           Revised October 2008


A copy of the book and the movie is available for US$100 plus shipping. Please email me at the address below for the shipping costs to your country from Canada. You may pay by sending a personal cheque in your local currency at the current rate of exchange or use PayPal with the email address below.

Serge Gracovetsky

209 Dauphine

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