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First published online July 25, 2005
Journal of Experimental Biology 208, 2831-2843 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01709

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A modified Hill muscle model that predicts muscle power output and efficiency during sinusoidal length changes

G. A. Lichtwark¹ and A. M. Wilson^1,2,*

¹ Structure and Motion Laboratory, Institute of Orthopaedics and Musculoskeletal Sciences, University College London, Royal National Orthopedic Hospital, Brockley Hill, Stanmore, Middlesex, HA7 4LP, UK
² Structure and Motion Laboratory, The Royal Veterinary College, Hawkshead Lane, North Mymms, Hatfield, Hertfordshire, AL9 7TA, UK

View larger version (22K): [in a new window]   Fig. 1. Properties of the muscle and definition of terms used in the model. These Scyliorhinus canicula white myotomal muscle properties were determined experimentally in the work of Curtin et al. (1998). (A) Force–velocity relationship of the contractile component at different levels of activation. (B) Variation in the SEE stiffness with force and (C) the resulting force–length relationship of the SEE. (D) Phase of activation is defined as the time between the start of stimulation and the start of shortening expressed as a percentage of cycle duration and is demonstrated with respect to one cycle of length change (a negative value corresponds to activating the muscle whilst the muscle is stretching). Duty cycle is expressed as the fraction of the cycle that the muscle/model is stimulated.

View larger version (20K): [in a new window]   Fig. 2. The rise and fall of the calcium transients (green) and the resulting activation level (blue) are shown for the original block model (A) and the twitch model (B). Twitches were applied to the model at a frequency of 22 Hz, the same as in the experimental design, and each individual stimulus (twitch) lasted 7.5 ms. The constants 1 (0.03), 2 (0.10), K (0.19) and n (2.8) were optimised to produce fits to respond to a single stimulus condition (Fig. 3). A time constant, (d=0.015 s), was also required to delay the onset of activation, as was seen in the experimental results. This time constant is thought to be due to the time taken for ATP turnover to proceed and for the calcium signalling to cause a calcium influx.

View larger version (17K): [in a new window]   Fig. 3. Comparison of output of model with that of the experimental results for the single optimised condition (frequency=1.25, duty cycle=0.121, stimulus phase=5). The experimental results are taken from raw data used in Curtin and Woledge (1996). (A) Length trajectory (relative to Lo) of MTU for the model (dotted line) and experimental results (solid line); (B) Force (relative to Po) output for the model (dotted line) and experimental results (solid blue line); (C) Energy output (heat + work) for the model (blue dotted line) and experimental results (blue solid line). The experimental results show small amplitude fluctuations as a result of heat measurement from thermopile. The experimental force recordings (solid blue line in B) and the estimated activation level (red line in D) were used as inputs into the energetic model to approximate energetic output using the experimental results (red line in C). (D) Activation level (the relative number of attached crossbridges) predicted by the model (dotted blue line), estimated from the experimental results (red line) and the stimulation pattern from the experiment (solid blue line). (E) CE velocity (Lo s–1) as predicted by the model (dotted blue line), and as estimated from the experimental results (solid blue line). This is shown in reference to the MTU velocity (red line).

View larger version (26K): [in a new window]   Fig. 4. Comparison of the model output to the experimental results across a range of stimulation conditions with activation constants optimised to fit a single stimulus condition (Fig. 3). Length, force, energetic output, activation and contractile component velocity are compared between the model output (dotted blue lines) and the experimental results (solid blue lines). The convention for these figures is the same as for the single condition in Fig. 3, with the red lines indicating the modelled energetic output for the experimental data (energy plot), the estimated experimental activation level (activation plot) and the MTU velocity [contractile component (CE) velocity plot]. The comparison is made at three different cycle frequencies: (A) 5 Hz, (B) 1.25 Hz and (C) 0.71 Hz, with varying duty cycle and phase.

View larger version (19K): [in a new window]   Fig. 5. The relationship between the activator (Ca2+) concentration and the activation level (the relative number of attached crossbridges) for each of the optimised values of K and n that produced best fits to the force data at each of the three cycle frequencies (5, 1.25 and 0.71 Hz).

View larger version (27K): [in a new window]   Fig. 6. Comparison of the model output to the experimental results across a range of stimulation conditions with activation constants optimised to best fit each individual cycle frequency (Fig. 3). Length, force, energetic output, activation and contractile component velocity are compared between the model output (dotted blue lines) and the experimental results (solid blue lines). The convention for these figures is the same as for the single condition in Fig. 3, with the red lines indicating the modelled energetic output for the experimental data (energy plot), the estimated experimental activation level (activation plot) and the MTU velocity [contractile component (CE) velocity plot]. The comparison is made at the frequencies of (A) 5 Hz and (B) 0.71 Hz.

View larger version (17K): [in a new window]   Fig. 7. Comparison of the experimental (blue) power output (A) and efficiency (B) to the model predictions (green) for a range of cycle frequencies and increasing duty cycle. The same phase of activation was used in each condition. Model results use the optimal activation parameters for each cycle frequency, as shown in Table 1.Power is defined as the work per cycle time and is scaled up from PoLo/cycle time units based on the properties of the muscle reported by Curtin et al. (1996).

View larger version (19K): [in a new window]   Fig. 8. (A) Comparison of the model (dotted lines) to individual records of length, force, energetic output and stimulation/activation for an example soleus muscle of a mouse (solid lines) as reported in Barclay (1994). Values are scaled according to average data reported by Barclay and personal communication. (B) Comparison of model (dotted lines) to experimental (solid lines) power and rate of heat and total energy output (power + heat rate) across a range of frequencies. The stimulation rate constants used were 1=0.045 and 2=0.045 and Vmax=4.0 Lo s–1.

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