unknown user

03-12-2000, 09:33 PM

Dear Fellow-Biomechanists,

On 3 March, Ton van den Bogert initiated a discussion on the above

topic within the TGCS-group. In my opinion, this problem is of

general interest to all biomechanists so that I have taken the

liberty to continue the discussion on the broader BIOMCH-L-forum.

In short, Ton posed the question as to the importance of including

the dependence of the maximum shortening velocity Vmax on

contractile element length and activation level into a muscle model.

Replies to Ton's question came from Peter Sinclair, A.J. van Soest,

and Jim Dowling. Peter Sinclair thinks this question is of importance

(model behavior in unloaded conditions), A.J. van Soest reduces the

question to one that can be solved by simulation (optimization) only,

while Jim Dowling relates to basic questions of modeling methodology

and to the experimental findings of profoundly nonlinear myodynamic

behavior by Peter Huijing (whose work I highly value) and his group.

Having observed these discussions for some time now I would like to

express my own opinion on this issue.

1. The DEPENDENCY OF THE MAXIMUM SHORTENING VELOCITY Vmax

on the length x of the contractile element and the active state q of

a muscle fiber or a motor unit (I am deliberately not saying

"muscle") is indeed of paramount importance to muscle modeling, and

therefore has always been included in my own muscle models. Ton van

den Bogert, unfortunately, completely misunderstood my muscle model

published in 1977, when he asserts in his mail of 3 March that I (and

others) had "simply scaled Hill's force-velocity curve by a

normalized force-length relationship and a normalized activation

level", with the (alleged) consequence "that Vmax does not depend on

length and activation".

In fact, the contrary is true. I have never used Hill's model of the

force-velocity curve but developed my own VELOCITY DEPENDENCE

FUNCTION of the force production of the contractile proteins from a

basic theory of molecular contraction published in 1973 (Hatze, H.

(1973) A theory of contraction and a mathematical model of striated

muscle. J. theor. Biol. 40, 219 - 246). A combination of this highly

nonlinear velocity dependence function with the filamentary overlap

and active state functions then yields the final CONTRACTION

DYNAMICAL DIFFERENTIAL EQUATION for the velocity dx/dt of the

contractile element. In this differential equation appears Vmax. If

this differential equation is solved for the special case of zero

force (unloaded shortening), a nonlinear relation results, predicting

Vmax as a decreasing function of both the contractile element length

x (for x below the optimum length) and the active state q. This is

described in detail on page 112 of: Hatze, H. (1977) A Myocybernetic

Control Model of Skeletal Muscle. Biol. Cybernetics 25, 103 - 119.

In this publication, the dependence of Vmax on the active state q

for optimal filamentary overlap (x = 1), and on the length x for

maximum active state (q = 1) are depicted in Fig. 6 and shown to

agree well with experimental data of Jewell and Wilkie (1960), and

Bahler et al. (1968). In addition, the predictions of my model also

agree well with the recent findings of Chow and Darling (1999) as

quoted by Ton in his mail: at active state q = 0.2 (20 %), Vmax was

found by Chow and Darling to drop to 60 % of its value at max.

activation, while my model predicts a drop to 66 % (Fig. 6, left

part, of Hatze, 1977) which agrees obviously quite well with the

experimental data.

The important point to note is that the functional dependence of Vmax

on q and x has not been included artificially in my model but follows

naturally from the model development based on a molecular contraction

theory. Many muscle models developed subsequently by others made use

of the basic elements of my myocybernetic model published in 1977.

I conclude this point by giving my humble opinion: the

Vmax-dependence on the active state and the contractile element

length is important and should be intrinsic in any appropirate muscle

model because it represents a real physiological phenomenon.

2. In muscle modeling it is frequently overlooked that a muscle

consists of MOTOR UNITS which represent the smallest neurally

distinctly controlable muscle entities. In general, the fibers of

different motor units belong to different fiber types, all of which

possess VASTLY VARYING CONTRACTILE PROPERTIES (twitch contraction

and half-relaxation times, activation constants, max. shortening

velocities, etc.). This is the reason, why in the beginning of the

discussion of point 1 above I deliberately avoided the term muscle.

There is obviously a good reason why nature devised a contractile

system comprising distinctly different motor units. (An attempt to

explain this is contained in: Hatze, H. (1979) A teleological

explanation of Weber's law and the motor unit size law. Bull. of

Mathem. Biology 41, 407 - 425). Thus, during the recruitment (or de-

recruitment) of motor units, the contractile properties of the whole

muscle change continously in a highly non-linear fashion, a

phenomenon which must also be included in any appropriate muscle

model. In my muscle model, this fact is accounted for by

subdividing the model into nine different motor units of

exponentially growing size and different contractile properties. Each

of these nine motor units is, of course, individually controlable by

a separate neural stimulation rate, only the contraction dynamics is

common to all motor units.

3. It should also be clear that any proper muscle model must account

for the phenomena relating to the HISTORY DEPENDANCE OF THE

CONTRACTILE PROCESSES (force enhancement during stretch and

potentiation after stretch, force depression after shortening, etc.).

I have dealt with these questions and how to model these functions in

my book: Myocybernetic Control Models of Skeletal Muscle.

4. My conclusive PERSONAL OPINION ON PROPER MUSCLE MODELING is

this: Any really appropriate muscle model must be capable of

predicting, within certain limits of accuracy, all major phenomena of

the contractile process that are known from experimental evidence.

Model simplifications may work for some simulation purposes but may

produce disastrous results in other applications, possibly without

the user even becoming aware of it.

5. Finally, I would like to express some personal thoughts on

expected FUTURE TRENDS IN GENERAL MODELING OF THE HUMAN

NEUROMYOSKELETAL SYSTEM. My feeling is that we need new and much

more sophisticated approaches resulting in a new generation of

models. First, the neural hyposensitivity phenomenon (large

perturbations of neural controls produce minimal deviations in

skeletal motion trajectories) and the ill-posedness of the associated

inverse dynamics problem will force us to rethink the validity of our

optimization models. (The problem is treated in Hatze, H. (2000) The

inverse dynamics problem of neuromuscular control. Biol. Cybernetics

82, 133 - 141). Secondly, we will probably have to discard

deterministic models in favour of stochastic and

biologically more realistic differential models. Thirdly, the

segmented rigid body models used for the skeletal system turn out to

be increasingly problematic when verifyable responses of the real

biosystem (the human limb system) are compared with model results,

particularly in the field of motion analysis (i.e., inverse

dynamics). They will have to be replaced by semi-amorphic hybrid

models of the skeletal system containing many elastic substructures

that are themselves muscle-state dependent.

There is a huge task ahead of us in the near future. I, and probably

others, have already begun working on it.

Herbert Hatze

Prof.Dr. Herbert Hatze

Head, Department and Laboratory of Biomechanics

ISW, University of Vienna

Auf der Schmelz 6 Tel.: +43-1-4277-48880

A-1150 WIEN FAX : +43-1-4277-48889

AUSTRIA e-mail: herbert.hatze@univie.ac.at

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On 3 March, Ton van den Bogert initiated a discussion on the above

topic within the TGCS-group. In my opinion, this problem is of

general interest to all biomechanists so that I have taken the

liberty to continue the discussion on the broader BIOMCH-L-forum.

In short, Ton posed the question as to the importance of including

the dependence of the maximum shortening velocity Vmax on

contractile element length and activation level into a muscle model.

Replies to Ton's question came from Peter Sinclair, A.J. van Soest,

and Jim Dowling. Peter Sinclair thinks this question is of importance

(model behavior in unloaded conditions), A.J. van Soest reduces the

question to one that can be solved by simulation (optimization) only,

while Jim Dowling relates to basic questions of modeling methodology

and to the experimental findings of profoundly nonlinear myodynamic

behavior by Peter Huijing (whose work I highly value) and his group.

Having observed these discussions for some time now I would like to

express my own opinion on this issue.

1. The DEPENDENCY OF THE MAXIMUM SHORTENING VELOCITY Vmax

on the length x of the contractile element and the active state q of

a muscle fiber or a motor unit (I am deliberately not saying

"muscle") is indeed of paramount importance to muscle modeling, and

therefore has always been included in my own muscle models. Ton van

den Bogert, unfortunately, completely misunderstood my muscle model

published in 1977, when he asserts in his mail of 3 March that I (and

others) had "simply scaled Hill's force-velocity curve by a

normalized force-length relationship and a normalized activation

level", with the (alleged) consequence "that Vmax does not depend on

length and activation".

In fact, the contrary is true. I have never used Hill's model of the

force-velocity curve but developed my own VELOCITY DEPENDENCE

FUNCTION of the force production of the contractile proteins from a

basic theory of molecular contraction published in 1973 (Hatze, H.

(1973) A theory of contraction and a mathematical model of striated

muscle. J. theor. Biol. 40, 219 - 246). A combination of this highly

nonlinear velocity dependence function with the filamentary overlap

and active state functions then yields the final CONTRACTION

DYNAMICAL DIFFERENTIAL EQUATION for the velocity dx/dt of the

contractile element. In this differential equation appears Vmax. If

this differential equation is solved for the special case of zero

force (unloaded shortening), a nonlinear relation results, predicting

Vmax as a decreasing function of both the contractile element length

x (for x below the optimum length) and the active state q. This is

described in detail on page 112 of: Hatze, H. (1977) A Myocybernetic

Control Model of Skeletal Muscle. Biol. Cybernetics 25, 103 - 119.

In this publication, the dependence of Vmax on the active state q

for optimal filamentary overlap (x = 1), and on the length x for

maximum active state (q = 1) are depicted in Fig. 6 and shown to

agree well with experimental data of Jewell and Wilkie (1960), and

Bahler et al. (1968). In addition, the predictions of my model also

agree well with the recent findings of Chow and Darling (1999) as

quoted by Ton in his mail: at active state q = 0.2 (20 %), Vmax was

found by Chow and Darling to drop to 60 % of its value at max.

activation, while my model predicts a drop to 66 % (Fig. 6, left

part, of Hatze, 1977) which agrees obviously quite well with the

experimental data.

The important point to note is that the functional dependence of Vmax

on q and x has not been included artificially in my model but follows

naturally from the model development based on a molecular contraction

theory. Many muscle models developed subsequently by others made use

of the basic elements of my myocybernetic model published in 1977.

I conclude this point by giving my humble opinion: the

Vmax-dependence on the active state and the contractile element

length is important and should be intrinsic in any appropirate muscle

model because it represents a real physiological phenomenon.

2. In muscle modeling it is frequently overlooked that a muscle

consists of MOTOR UNITS which represent the smallest neurally

distinctly controlable muscle entities. In general, the fibers of

different motor units belong to different fiber types, all of which

possess VASTLY VARYING CONTRACTILE PROPERTIES (twitch contraction

and half-relaxation times, activation constants, max. shortening

velocities, etc.). This is the reason, why in the beginning of the

discussion of point 1 above I deliberately avoided the term muscle.

There is obviously a good reason why nature devised a contractile

system comprising distinctly different motor units. (An attempt to

explain this is contained in: Hatze, H. (1979) A teleological

explanation of Weber's law and the motor unit size law. Bull. of

Mathem. Biology 41, 407 - 425). Thus, during the recruitment (or de-

recruitment) of motor units, the contractile properties of the whole

muscle change continously in a highly non-linear fashion, a

phenomenon which must also be included in any appropriate muscle

model. In my muscle model, this fact is accounted for by

subdividing the model into nine different motor units of

exponentially growing size and different contractile properties. Each

of these nine motor units is, of course, individually controlable by

a separate neural stimulation rate, only the contraction dynamics is

common to all motor units.

3. It should also be clear that any proper muscle model must account

for the phenomena relating to the HISTORY DEPENDANCE OF THE

CONTRACTILE PROCESSES (force enhancement during stretch and

potentiation after stretch, force depression after shortening, etc.).

I have dealt with these questions and how to model these functions in

my book: Myocybernetic Control Models of Skeletal Muscle.

4. My conclusive PERSONAL OPINION ON PROPER MUSCLE MODELING is

this: Any really appropriate muscle model must be capable of

predicting, within certain limits of accuracy, all major phenomena of

the contractile process that are known from experimental evidence.

Model simplifications may work for some simulation purposes but may

produce disastrous results in other applications, possibly without

the user even becoming aware of it.

5. Finally, I would like to express some personal thoughts on

expected FUTURE TRENDS IN GENERAL MODELING OF THE HUMAN

NEUROMYOSKELETAL SYSTEM. My feeling is that we need new and much

more sophisticated approaches resulting in a new generation of

models. First, the neural hyposensitivity phenomenon (large

perturbations of neural controls produce minimal deviations in

skeletal motion trajectories) and the ill-posedness of the associated

inverse dynamics problem will force us to rethink the validity of our

optimization models. (The problem is treated in Hatze, H. (2000) The

inverse dynamics problem of neuromuscular control. Biol. Cybernetics

82, 133 - 141). Secondly, we will probably have to discard

deterministic models in favour of stochastic and

biologically more realistic differential models. Thirdly, the

segmented rigid body models used for the skeletal system turn out to

be increasingly problematic when verifyable responses of the real

biosystem (the human limb system) are compared with model results,

particularly in the field of motion analysis (i.e., inverse

dynamics). They will have to be replaced by semi-amorphic hybrid

models of the skeletal system containing many elastic substructures

that are themselves muscle-state dependent.

There is a huge task ahead of us in the near future. I, and probably

others, have already begun working on it.

Herbert Hatze

Prof.Dr. Herbert Hatze

Head, Department and Laboratory of Biomechanics

ISW, University of Vienna

Auf der Schmelz 6 Tel.: +43-1-4277-48880

A-1150 WIEN FAX : +43-1-4277-48889

AUSTRIA e-mail: herbert.hatze@univie.ac.at

---------------------------------------------------------------

To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl

For information and archives: http://isb.ri.ccf.org/biomch-l

---------------------------------------------------------------