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The mechanisms underlying many important properties of the human atrial action potential (AP) are poorly understood. Using specific formulations of the K+, Na+, and Ca2+ currents based on data recorded from human atrial myocytes, along with representations of pump, exchange, and background currents, we developed a mathematical model of the AP. The model AP resembles APs recorded from human atrial samples and responds to rate changes, L-type Ca2+ current blockade, Na+/Ca2+ exchanger inhibition, and variations in transient outward current amplitude in a fashion similar to experimental recordings. Rate-dependent adaptation of AP duration, an important determinant of susceptibility to atrial fibrillation, was attributable to incomplete L-type Ca2+ current recovery from inactivation and incomplete delayed rectifier current deactivation at rapid rates. Experimental observations of variable AP morphology could be accounted for by changes in transient outward current density, as suggested experimentally. We conclude that this mathematical model of the human atrial AP reproduces a variety of observed AP behaviors and provides insights into the mechanisms of clinically important AP properties.
- action potential morphology
- action potential rate dependence
- transient outward current
- L-type calcium current
- sodium/calcium exchanger
the human atrial action potential (AP) and ionic currents that underlie its morphology are of critical importance to our understanding of the electrical properties of atrial tissues in normal and pathological conditions. AP shape and underlying ionic current densities are frequency dependent and may be modulated by specific pharmacological agents (33). Their properties are known to change during the course of atrial arrhythmias or as a result of other cardiac pathologies (32, 39). In addition, APs have shown significant variability in morphology, ranging from triangular APs with no sustained plateaus to long APs with definite spike-and-dome morphology, even when recorded from cells isolated from the same atrial sample (4,33, 54). There is a need to integrate the information obtained from measurements of single currents to understand complex current interactions during the AP and their role in controlling AP morphology.
Until recently, limited data were available on the various currents underlying the AP. New findings have revealed important properties and significant interspecies differences. For example, the transient outward current (I to), present in human and rabbit atrial cells (8, 16, 25, 47), has been shown to recover from inactivation at least two orders of magnitude faster in humans than in rabbits (20, 47). A novel delayed rectifier current (I Kur) has been identified in human atrial cells (1, 18, 41, 48, 53) and is different from the sustained current (I sus) observed in rabbit atrial cells (12).I Kur also differs from delayed rectifiers in other species that show kinetically similar characteristics (3, 6, 29). Additional data have been obtained on the classic delayed rectifier currents (I Kr and I Ks) (54,55), the time-independent inward rectifier current (I K1) (30, 31), the fast Na+ current (I Na) (42), the L-type Ca2+ current (I Ca,L) (35) and its inactivation by voltage and intracellular Ca2+ (26, 49), and the Na+/Ca2+ exchanger current (I NaCa) (4).
Models of atrial cells based on animal data only have been published (14, 36, 40, 60). Although these models have provided valuable insights into the mechanisms underlying AP generation in these species, the existence of significant interspecies differences and the amount of human data available indicate a need for an AP model based specifically on direct measurements of human atrial currents. To address this issue, we developed a mathematical model of the AP based on ionic current data obtained directly in human atrial cells. When human data were insufficient to characterize a given atrial current fully, we supplemented with animal data and referred to existing published models of atrial and ventricular APs (14, 36, 37, 40, 60). The most recent published models of mammalian cardiac cells are those of Luo and Rudy (37) (LR2 model) and Linblad et al. (36) (LMCG model), based on measurements from guinea pig ventricular cells and rabbit atrial cells, respectively. Our model builds mostly on the work of Luo and Rudy to develop a working model of the human atrial AP.
Our primary goal was to develop a useful model of the AP from which we could gain insights into experimental observations made on human atrial cells and tissues and make predictions regarding the behavior of these cells under previously untested conditions. Throughout our description of the model, we clearly indicate the sources of data used to derive model parameters, how the parameters were obtained, and whether they were derived directly from data or modified to improve our representation of the AP. We then use the model to investigate the mechanism of experimentally measured AP rate dependence, the changes in AP morphology in the presence of pharmaceutical blockers of theI Ca,L and the Na+/Ca2+exchanger, and the variability in experimentally observed AP morphology.
- Transmembrane potential
- Gas constant
- Temperature
- Faraday constant
- Cm
- Membrane capacitance
- Human atrial action potential
- V˙max
- Maximal AP upstroke velocity (dV/dtmax)
- AP amplitude (peak AP voltage minus diastolic voltage at onset of AP)
- AP overshoot (peak AP voltage >0 mV)
- AP duration
- APD measured from AP onset to 50% of total APA repolarization
- APD measured from AP onset to 90% of total APA repolarization
- Atrial fibrillation
- Current-voltage
- Temperature adjustment factor, k(T ) =k(T0)
Sarcoplasmic reticulum
Junctional SR, SR release compartment
Network SR, SR uptake compartment
Cell volume
Intracellular volume
SR uptake compartment volume
SR release compartment volume
[X ]o
Extracellular concentration of ion X
[X ]i
Intracellular concentration of ion X
Calmodulin, sarcoplasmic Ca2+ buffer
Troponin, sarcoplasmic Ca2+ buffer
Calsequestrin, JSR Ca2+ buffer
Ca2+ concentration in release compartment
Ca2+ concentration in uptake compartment
Ca2+-bound calmodulin concentration
Ca2+-bound troponin concentration
Ca2+-bound calsequestrin concentration
Equilibrium potential for ion X
I ion
Total ionic current
I st
Stimulus current
Forward rate constant for gating variable x
Backward rate constant for gating variable x
Time constant for gating variable x
x ∞
Steady-state relation for gating variable x
I Na
Fast inward Na+ current
g Na
Maximal I Na conductance
Activation gating variable for I Na
Fast inactivation gating variable for I Na
Slow inactivation gating variable for I Na
Inward rectifier K+ current
g K1
Maximal I K1 conductance
I to
Transient outward K+ current
g to
Maximal I to conductance
Q10-based temperature adjustment factor
o a
Activation gating variable for I to
o i
Inactivation gating variable for I to
I Kur
Ultrarapid delayed rectifier K+ current
g Kur
Maximal I Kur conductance
u a
Activation gating variable for I Kur
u i
Inactivation gating variable for I Kur
I Kr
Rapid delayed rectifier K+ current
g Kr
Maximal I Kr conductance
x r
Activation gating variable for I Kur
I Ks
Slow delayed rectifier K+ current
g Ks
Maximal I Ks conductance
x s
Activation gating variable for I Ks
I Ca,L
L-type inward Ca2+ current
g Ca,L
Maximal I Ca,L conductance
Activation gating variable for I Ca,L
Voltage-dependent inactivation gating variable forI Ca,L
f Ca
Ca2+-dependent inactivation gating variable forI Ca,L
I p,Ca
Sarcoplasmic Ca2+ pump current
I p,Ca(max)
Maximal I p,Ca
I NaK
Na+-K+ pump current
I NaK(max)
Maximal I NaK
f NaK
Voltage-dependence parameter for I NaK
[Na+]o-dependence parameter forI NaK
K m,Na(i)
[Na+]i half-saturation constant forI NaK
K m,K(o)
[K+]o half-saturation constant forI NaK
I NaCa
Na+/Ca2+ exchanger current
I NaCa(max)
I NaCa scaling factor
K m,Na
[Na+]o saturation constant forI NaCa
K m,Ca
[Ca2+]o saturation constant forI NaCa
k sat
Saturation factor for I NaCa
Voltage-dependence parameter for I NaCa
I b,Na
Background Na+ current
g b,Na
Maximal I b,Na conductance
I b,Ca
Background Ca2+ current
g b,Ca
Maximal I b,Ca conductance
I rel
Ca2+ release current from the JSR
k rel
Maximal Ca2+ release rate for I rel
Activation gating variable for I rel
Ca2+ flux-dependent inactivation gating variable forI rel
Voltage-dependent inactivation gating variable forI rel
Sarcoplasmic Ca2+ flux signal for I rel
I up
Ca2+ uptake current into the NSR
I up(max)
Maximal Ca2+ uptake rate forI up
Maximal Ca2+ concentration in NSR
I tr
Ca2+ transfer current from NSR to JSR
Ca2+ transfer time constant
I up,leak
Ca2+ leak current from the NSR
Total calmodulin concentration in myoplasm
Total troponin concentration in myoplasm
Total calsequestrin concentration in JSR
K m,Cmdn
Ca2+ half-saturation constant for calmodulin
K m,Trpn
Ca2+ half-saturation constant for troponin
K m,Csqn
Ca2+ half-saturation constant for calsequestrin
V rest
Resting membrane potential
We model the cell membrane as a capacitor connected in parallel with variable resistances and batteries representing the ionic channels and driving forces. The time derivative of the membrane potential V(with the assumption of an equipotential cell) is given by Equation 1where I ion and I stare the total ionic current and stimulus current flowing across the membrane and C m is the total membrane capacitance. The total ionic current is given by Equation 2The ionic and pump currents, along with the handling of intracellular Ca2+ concentration ([Ca2+]i) by the sarcoplasmic reticulum (SR) system, are described in Membrane Currents. The model keeps track of the [Ca2+]i as well as intracellular concentrations of Na+([Na+]i) and K+([K+]i). No extracellular cleft spaces are included in the model (extracellular ion concentrations are fixed). Figure 1 shows a schematic representation of the currents, pumps, and exchangers included in the model.C m is taken to be 100 pF. The length and diameter of the cell are set to 100 and 16 μm, respectively. Unless otherwise noted, physical units are as follows: time (t) is in milliseconds, V is in millivolts, C m is in picofarads, current density is in picoamperes per picofarad, conductance is in nanosiemens per picofarad, and concentrations are in millimoles per liter. Physical parameters for the model cell are given in Table 1. Cell compartment volumes are identical to those used by Luo and Rudy (37).
A modified Euler method is used to integrate Eq. 1 with a fixed time step Δt = 0.005 ms (see , Numerical Integration). All simulations were performed on a local workstation (SGI Indigo 2XZ R4400) using double-precision arithmetic. A description of the model current formulations and their general mathematical representations are given below. A detailed listing of all equations and parameters can be found in the .
Membrane Currents
Fast Na+ current.
The model implements I Na as a modification of the widely used Ebihara-Johnson model (15) proposed by Luo and Rudy (37). The current is given by Equation 3Using the formulation of Luo and Rudy, we find that the model steady-state inactivation and its time constants are shifted 20 mV positive compared with the data of Sakakibara et al. (42). However, the formulation for steady-state inactivation is consistent with more recent human atrial steady-state inactivation data from Schneider et al. (46) (see discussion for a detailed evaluation of available human atrial data). Maximum Na+ conductance (g Na) was adjusted to produce an appropriate value for the AP amplitude (APA) and maximal upstroke velocity (V˙ max). We useg Na = 7.8 nS/pF, which is ∼1.3 times the temperature-adjusted value (with the assumption of Q10 = 3) reported by Sakakibara et al. Figure2 shows the steady-state activation and inactivation curves, as well as time constants, used in the model. Data from Sakakibara et al., shifted by the amounts described above and temperature corrected, are overlaid on the model curves. For activation time constants, temperature-adjusted data from Schneider et al. are shown. The peak current-voltage (I-V ) relationship for I Na is displayed in Fig.3. No experimental I-Vcurve was included, because available data (19, 42, 46), as a result of equal intra- and extracellular Na+ concentrations, display a nonphysiological reversal potential for I Na close to 0 mV.
Inward rectifier K+ current.
The model implements I K1 on the basis of available current data and resting membrane resistance measurements. The current is given by Equation 4We assume no temperature dependence of the inward rectifier current on the basis of preliminary experimental data. The current is assumed to reverse at E K, with a voltage-dependent conductance and no additional dependence on outward K+ concentration ([K+]o). Koumi et al. (30, 31) reported measurements of I K1from human atrial cells. When incorporated in the model, their large conductance value of 0.25 nS/pF at E K yielded a low input membrane resistance (∼40 MΩ for a 100-pF cell). We decreasedI K1 conductance to compensate for this using g K1 = 0.09 nS/pF. When the latter is used, input resistance calculated for a voltage step from −80 to −90 mV is ∼174 MΩ, closer to experimentally recorded values (e.g., ∼150 MΩ in Ref. 52). However, there is a large range of measured values for the input resistance of human atrial cells that may reflect differences in the isolation procedure and its effect on isolation-sensitive currents such as I K1. TheI-V relationship of I K1 is shown in Fig. 4.
Transient outward and ultrarapid rectifier K+currents.
The model implements I to andI Kur on the basis of data from our laboratory (20,53). The currents are given by Equation 5 Equation 6Both currents share similar activation kinetics but differ in their inactivation properties. I to has fairly rapid inactivation kinetics; I Kur displays only partial slow inactivation. Currents were modeled at room temperature, and Q10 = 2.2 was used to adjust their kinetics to 37°C (53). Current amplitudes were adjusted to match AP morphology. Figure5 shows steady-state activation, inactivation, and corresponding time constants forI Kur, along with experimental data from our laboratory (53). Figure 5 shows similar results forI to. All model fits shown in Fig. 5 are shifted 10 mV negative for computations to account for the known positive shift (13, 28) induced by divalent cations such as Cd2+, which was used experimentally to block I Ca,L. The peakI-V relationship of both currents, along with scaled experimental I-V curves from Feng et al. (19), are shown in Fig. 3. Figure 6 displays traces for I Kur and I to in response to voltage-clamp steps from a holding potential of −80 mV. The traces ofI to + I Kur show typical rapid inactivation of I to followed by a sustained outward current carried by I Kur. The combined model current traces are overlaid with scaled experimental recordings from Wang et al. (53).
Rapid and slow delayed rectifier K+ currents.
The model implements the two components of the classic delayed rectifier, I Kr and I Ks, on the basis of data from our laboratory (54, 55). The currents are given by Equation 7 Equation 8Both currents have only a single activation gate, althoughI Ks activation uses a squared (x 2) activation gate, as suggested in previous studies (61). Partial inactivation of I Kr is observed in some experiments (55) but is not included in this model. Figure 7 shows the steady-state activation curves and time constants for both delayed rectifier currents as well as experimentally measured time constants. The steady-state activation parameters are taken directly from fits to experimental data presented by Wang et al. (55). Some experimental evidence (27, 43, 45) suggests that activation and deactivation of I Ks may be multiexponential. To account for these measurements and obtain results consistent with the overall experimental literature, the model computations use time constants for I Ks activation that are one-half the fitted values shown in Fig. 7. In addition, observations of slow deactivation of the delayed rectifier at negative potentials (27, 43) were considered by using asymmetric relationships to represent the dependence of τx(r) and τx(s) on V. Current amplitudes were adjusted to match AP morphology. The I-V relationships of both currents, along with scaled experimental I-Vcurves from Wang et al., are shown in Fig. 3. Figure8 displays traces forI Kr and I Ks in response to 3,000-ms voltage-clamp steps from a holding potential of −80 mV. The model traces are overlaid with scaled experimental recordings from Wang et al. The combined model I K trace shows rapid and slow activation phases linked to activation of I Krand I Ks, respectively.
L-type Ca2+ current.
The model implements the slow inward Ca2+ current on the basis of previous models and data from human atrial cells (35, 37, 40,49). The current is given by Equation 9Our formulation follows that of Luo and Rudy (37) and includes voltage-dependent activation and inactivation gates, as well as a Ca2+-dependent inactivation gate. The steady-state activation relation was fit to human atrial data from Li and Nattel (35) (Fig. 9) and hence differs slightly from that of Luo and Rudy and Rasmusson et al. (40) by providing a better fit to the slower onset of activation observed in the experimental data. Because of the absence of activation time constant measurements from human atrial myocytes, we retained the activation time constant formulation reported previously (36, 37, 40). Steady-state voltage-dependent inactivation data from Li and Nattel were accurately fit by the relation of Rasmusson et al. (40) (Fig. 9). Hence, we retained their formulation in our model. Our formulation of steady-state Ca2+-dependent inactivation is slightly modified from that of Luo and Rudy to better match the Ca2+transient and reproduce the extent of Ca2+-dependent inactivation of I Ca,L observed experimentally. A fixed intrinsic time dependence (τf,Ca = 2 ms) is incorporated into the f Ca gate, as reported by Friedman et al. (21, 22) (see SR Ca2+ storage and release), to better reproduce the time course of Ca2+-dependent inactivation of I Ca,L.
In experiments on human atrial cells, Sun et al. (49) showed two fast Ca2+-dependent inactivation time constants forI Ca,L, on the order of 10–40 and 300–400 ms, respectively, at room temperature. In addition, when monovalent cations or Sr2+ was used as charge carrier, a very slow voltage-dependent process was apparent with inactivation time constants in the range of 600–800 ms at room temperature. Our ability to reproduce these results is dependent on our formulation of the current, which implements two inactivation mechanisms forI Ca,L. We have selected the time constant for voltage-dependent inactivation in our model to match qualitatively the measurements of Sun et al. The voltage dependence of the time constant uses the form proposed by Luo and Rudy (37), but the magnitude of the time constant is scaled to better agree with the data of Sun et al. The time constant at 0 mV is ∼240 ms in the model, which is within the experimentally measured range with the assumption of a threefold temperature correction to 37°C. Ca2+-dependent inactivation is mediated directly through the f Cagate. We rely on the Ca2+ transient to drive Ca2+-dependent inactivation, with its time constants being set indirectly through the dynamics of Ca2+ release and reuptake. This provides rapid inactivation of I Ca,Lon a time scale on the order of 10 ms. The model does not account for the existence of a slower Ca2+-dependent inactivation, as identified by Sun et al. (see discussion). TheI-V relationships for I Ca,L, along with a scaled experimental I-V curve from Li and Nattel (35), are shown in Fig. 3. Figure10 A displays model traces forI Ca,L in response to voltage-clamp steps from a holding potential of −80 mV. The current traces exhibit rapid activation, followed by two distinct inactivation time scales. Rapid inactivation mediated by intracellular Ca2+ inactivates a large fraction of the current soon after the onset of the voltage step and is followed by a slower voltage-dependent inactivation. The model response after a voltage-clamp step to 0 mV is compared with a scaled averaged experimental recording from Sun et al. in Fig.10 B.
Na+-K+ pump.
Our formulation of the Na+-K+ pump follows the model of Luo and Rudy (37). The amplitude of the pump current [I NaK(max)] is adjusted to maintain stable intracellular ion concentrations at rest (37, 59). The pump current is given by Equation 10The voltage dependence of I NaK at resting [Na+]i([K+]o is fixed at 5.4 mM in the model) is shown in Fig. 4.
Na+/Ca2+ exchanger.
The exchanger current follows the revised formulation of Luo and Rudy (37). The exchanger current is given by
As with I NaK, the amplitude is adjusted to maintain stable physiological intracellular ion concentrations at rest (Table 2). Figure11 shows the voltage dependence of the exchanger current at resting [Na+]i for two values of [Ca2+]i: at rest ([Ca2+]i = 0.1 μM) and near peak Ca2+ release concentration ([Ca2+]i = 1 μM). The downward shift in the curve is associated with a shift from Ca2+ entry to Ca2+ extrusion mode at depolarized potentials as Ca2+ concentration increases.
Background Ca2+ and Na+ currents.
The Ca2+ leak current is adjusted to maintain a stable [Ca2+]i at rest. The Na+ leak current is adjusted along with I NaK to maintain stable [Na+]i and [K+]i at rest. Both approaches have been used extensively in previous AP models (37, 59). The background currents are given by Equation 12 Equation 13
Ca2+ pump current.
A sarcolemmal Ca2+ pump is included in the model to maintain [Ca2+]i at physiological levels. The pump current formulation (37, 40) is given by Equation 14Figure 4 shows the total time-independentI-V relationship for the model, which includes the background currents, I K1, and the pump and exchanger currents.
SR Ca2+ storage and release.
SR Ca2+ uptake and release are implemented using a two-compartment model, as reported by Luo and Rudy (37). Intracellular Ca2+ is taken up into an SR uptake compartment [network SR (NSR), 5.52% of cell volume] coupled to a release compartment [junctional SR (JSR), 0.48% of cell volume]. Uptake is [Ca2+]i dependent and includes a leak from the uptake compartment back into the intracellular space. The main currents (in mM/ms) involved in SR Ca2+ handling are given by Equation 15 Equation 16 Equation 17
Triggering of Ca2+ release from the JSR is modified from the original formulation of Luo and Rudy (37) following the work of Friedman et al. (21, 22). Release from the JSR is triggered by Ca2+ flux into the cell with emphasis placed on flux through the closely coupled I Ca,L and JSR release channels. Release is controlled by two Ca2+ flux-dependent activation and inactivation gates, u and v. Steady-state activation and inactivation curves for u andv, as well as their respective time constants, are shown in Fig. 12. This trigger mechanism results in a transient activation of the SR Ca2+ release current in response to a suprathreshold Ca2+ flux. Friedman (21) showed that, for AP simulations, this formulation is equivalent to using a threshold on [Ca2+]i entry, as done by Luo and Rudy (37). The formulation used here has the advantage of being a continuous analytic representation, thereby avoiding possible discontinuities and artifacts in the cellular response near the threshold for Ca2+ release (21). The voltage-dependent inactivation gate w contributes to the decrease in SR release current amplitude at positive potentials, when membrane voltage approaches the I Ca,L reversal potential. The steady-state voltage-dependent inactivation curve for w and its time constant are also shown in Fig. 12. Detailed expressions for the flux computations and gating variables are given in the . In a recent experimental study on human atrial myocytes, Hatem et al. (26) recorded Ca2+ transients in response to 150-ms voltage-clamp pulses to 0 mV applied at frequencies of 0.1 and 1 Hz. Figure 13 shows a comparison of experimental data from Hatem et al. and model-generated Ca2+ transients elicited using the same voltage-clamp protocol. A detailed comparison of model and experimental Ca2+ transient morphologies can be found in thediscussion.
Myoplasmic and SR Ca2+ buffers.
Ca2+ buffering within the cytoplasm is mediated by troponin and calmodulin. Ca2+ buffering in the release compartment is mediated by calsequestrin. In particular, calsequestrin is essential in providing sufficient Ca2+ stores for release within the limited JSR volume. Buffers are considered at equilibrium throughout our simulations, and incoming Ca2+in the myoplasm and JSR compartments is instantaneously equilibrated into free and buffer-bound fractions (see for computational methods). Buffer concentrations and binding constants are given in Table 1 and follow the models of Luo and Rudy (37) and Rasmusson et al. (40).
Resting Properties
The model has a stable resting potential near −81 mV. All intracellular concentrations are stable at rest with [Ca2+]i = 0.1 μM, [Na+]i = 11.2 mM, and [K+]i = 139.0 mM. As stated earlier, the resting input resistance measured as the current change from −80 to −90 mV is ∼174 MΩ. The first 100-ms period of Fig.14 shows the balance of membrane currents in diastole during stimulation at 1,000 ms. During the diastolic phase the steady state involves a balance between the pump and exchanger currents, the background currents, andI K1.
Model AP
Figure 14 shows a model AP generated during stimulation at 1,000 ms in response to 2-ms pulses of 2-nA amplitude (twice diastolic threshold). The time course of membrane currents during the AP is also shown in Fig. 14. AP duration (APD) at 50 and 90% repolarization (APD50 and APD90), APA, AP overshoot (APO), and V˙ max are reported in Table 3. The AP generated with the control parameters exhibits a spike-and-dome morphology commonly observed in human atrial AP recordings. This morphology is similar to experimentally recorded spike-and-dome APs reported by Benardeau et al. (4) and Wang et al. (53). Intracellular Ca2+ dynamics during the AP are shown in Fig. 15.
AP Rate Dependence
Rate dependence of APD and AP refractoriness is an essential property of atrial cells that is central to our understanding of excitability and propagation patterns in the atria, particularly during reentrant arrhythmias. Table 3 summarizes the main properties of the model APs at various pacing periods. Figure 16displays model APs, after 12 s of pacing, for various stimulation periods ranging from 1,000 to 300 ms. There is no significant increase in APD at >1,000 ms. This is in agreement with data from Fermini et al. (20) that showed no change in AP morphology and APD on changing the stimulation period abruptly from 10 to 1 s. Larger relative rate-dependent changes in APD are measured with APD50than with APD90, as in experimental preparations (20, 32). Figure 16 also shows APs obtained in a tissue preparation at stimulation periods of 1,000, 600, and 300 ms (56). The experimentally observed changes in APD and AP morphology with stimulation rate are qualitatively similar to those observed in the model.
The mechanism underlying APD changes may be investigated by examining the detailed current traces for APs at stimulation periods of 1,000 and 300 ms depicted in Fig. 16. The traces reveal that an increase in stimulation rate conspicuously reduces I Ca,L. We inspected the time course of gating variables forI Ca,L, I Kr, andI Ks to quantify their role in rate-dependent AP changes. We first consider a comparison of the slow gating variablesf, f Ca, x r, and at the onset of the AP for stimulation periods of 1,000 and 300 ms. These slow variables are most susceptible to alteration at fast rates. ForI Kr and I Ks, we found that incomplete deactivation at 300 ms resulted in an increase of 0.05 forx r and 0.004 for compared with their values at 1,000 ms. This compares with reductions of 0.2 for fand 0.16 for f Ca. The maximal conductances ofI Ca,L and I Ks are comparable, whereas the maximal conductance of I Kr is four times smaller. Hence, the changes in the f andf Ca gate result in larger changes in current at the faster rate, pointing to I Ca,L as an important determinant of AP rate dependence. However, the absolute magnitude of current changes with rate must be considered relative to the amplitude of other currents activating at a similar stage during the AP. For this reason, a small rate-dependent change in I Kamplitude may have significant effects during late repolarization, whenI K is the dominant repolarizing current.
To better ascertain the relative importance of I Kand I Ca,L in AP rate adaptation, we use a simulation protocol where the rate-induced changes in a specific current are identified and implemented individually to determine its specific contributions to AP rate dependence. We begin with a simulation of pacing at 1,000 ms that is interrupted just before the onset of a stimulus. Before continuation of the simulation and application of the scheduled stimulus, the values of the gating variables for the current under study are substituted with their values just before a stimulus during pacing at 300 ms. When the simulation is continued and the scheduled stimulus is applied, the resulting AP displays the rate-adaptation properties associated with the current under study, the state of which has been reset to its value during rapid pacing. We have carried out this procedure to examine the rate-dependent effect of I Ca,L (via the dand f gating variables), I Kr (via the x r gating variable), I Ks(via the x s gating variable), and [Ca2+]i (via [Ca2+]i, [Ca2+]up, [Ca2+]rel, and the u, v, and w gating variables) on the AP.
Figure 17 displays the APs after 12 s of pacing at 1,000 and 300 ms, along with the APs arising from isolating the rate-adaptation effects of I Ca,L,I K(I Kr + I Ks), andI Ca,L + I K. We note that when the effect of either I Kr or I Ksalone was examined, the observed change in APD90 was ∼50% of the total reduction seen with both currents combined (I K in Fig. 17). Also, the effect on APD90 of [Ca2+]i andI Ca,L combined was similar to that ofI Ca,L alone, which is why the results for [Ca2+]i were not included in Fig. 17. The limited role of [Ca2+]i at fast pacing rates may reflect the opposing effects of a higher resting [Ca2+]i and a lower Ca2+transient amplitude on I Ca,L inactivation. Our first observation is that the effect of the rate-dependent decrease in available I Ca,L alone is to lower the plateau potential and to significantly accelerate early repolarization. The lower plateau level reduces I K activation, slows terminal repolarization, and results in no net change in total APD. Second, we observe that the effect of the rate-dependent increase in available I K alone is to lower the plateau potential and to compensate for the effect of the reduced plateau level on I K activation, leaving the rate of terminal repolarization unchanged. The effect of I K rate adaptation alone explains less than one-half of the total rate-dependent reduction in APD90. Third, the effect ofI Ca,L and I K combined is synergistic, lowering the plateau potential and accelerating late repolarization relative to the effect of I Ca,Lalone. Changes in these two currents together are sufficient to account for the total rate-dependent AP shortening observed on reducing the pacing period from 1,000 to 300 ms. Hence, we conclude thatI Ca,L and I K act together to produce the observed rate-dependent AP shortening.
ICa,L and Rate Dependence
Li and Nattel (35) showed that rate dependence can be abolished in isolated human atrial myocytes by blocking I Ca,L. To compare these experimental data with the behavior of the model, we evaluated the effects of a strong decrease in L-type Ca2+channel conductance on model AP rate dependence. Figure18 A shows model APs after 12 s of pacing with stimulation at 5,000 and 300 ms with a 90% reduction inI Ca,L conductance. Figure 18 B shows corresponding experimental results from Li and Nattel in which nifedipine was used to block I Ca,L. In the model and experiments, block of I Ca,L abolishes AP rate dependence. Although we have shown that I Ca,L andI K play a role in rate adaptation, it appears that when I Ca,L is strongly inhibited, as in Fig. 18, the plateau level is lowered to the point where I Kactivation is greatly reduced and can no longer contribute to rate adaptation.
Block of INaCa and Role in APD
Recently, Benardeau et al. (4) investigated the role of Na+/Ca2+ exchange in AP morphology and APD. In particular, they examined the effect of I NaCa block by Li+ on AP morphology. Their results revealed a shortening of APD and transient hyperpolarization after the AP in the presence of Li+. Figure 19shows control and I NaCa-blocked (90% decrease in maximum exchanger activity compared with control) model APs with stimulation at 1 s. Model APs display both of the changes observed in experimental preparations (see Fig. 7A in Ref. 4), i.e., AP shortening and hyperpolarization. Inspection of the current records shows that the decrease in APD may be related to two mechanisms:1) an increase in Ca2+-dependentI Ca,L inactivation caused by an increase in resting [Ca2+]i and Ca2+ transient amplitude and 2) a reduction in depolarizing exchanger current during the late phase of the AP. The observed membrane hyperpolarization is due to a reduced inward exchanger current at rest, causing a negative shift of the resting membrane potential.
Variability in AP Morphology
Using the model, we attempted to simulate changes in current density that could explain the changes in human atrial AP morphology seen experimentally by many investigators (1, 4, 17, 24, 50, 51, 54). It has been suggested that variations in the relative density of the transient outward current may contribute to some of the AP variability (17, 54). We concentrated on three currents: I to,I Kur, and I Ca,L. Figure20 shows the result of varying the density of each of these three currents, from one-tenth to three times the control conductance value, on AP morphology. Figure 20,A–D, shows a continuum of AP morphologies after 12 s of pacing at a stimulation period of 1,000 ms for various modifications of the current densities. ForI Ca,L, we observe an increase in APD90 as I Ca,L conductance increases. There is little change in APD90 as I Kurconductance decreases. For I to, we observe an increase in APD90 as I to conductance decreases for large conductances, then a decrease in APD90as I to conductance decreases for small conductances. Of the three currents we have investigated, it appears that only the I to variations can explain the whole spectrum of AP morphologies observed experimentally (4, 33, 54). This ranges from rectangular APs with a short plateau at the lowestI to conductance values to shorter triangular APs at high I to conductance values, through long spike-and-dome morphologies at intermediate I toconductance values. Both I Kur andI Ca,L alone can explain a change from a spike-and-dome to a triangular AP shape, but a reduction inI to is required in the model to produce the high plateau and absence of notch characteristic of a rectangular AP (54). In addition, the relation between I to and AP morphology in the model parallels the relative changes inI to observed experimentally in the AP classification of Wang et al. (54).
We have developed a mathematical model of the human atrial AP. Whenever possible, we used data directly measured on human atrial cells to derive model parameters. When human data were not available to completely characterize a given current, we relied on existing AP models based mainly on guinea pig ventricular (37) and rabbit atrial (36) data. The model yields AP morphologies that are consistent with a variety of experimental observations and gives potential insights into their underlying ionic mechanisms.
Model Current Formulations Compared with Experimental Data
Fast Na+ current.
I Na is a large, rapid inward current that has proved difficult to measure accurately using voltage-clamp techniques. Data from isolated human atrial cells are available elsewhere (19, 42). However, these data for steady-state activation and especially inactivation of the current are shifted to negative potentials compared with other human (7, 46) and animal (57) data. According to the steady-state inactivation curves in these studies, ∼90% of the available I Na would be inactivated at resting potentials positive to −80 mV. In conjunction with the small maximum conductances measured in these human cells (40 pA/pF maximum current in Ref. 19 compared with 200 pA/pF in Ref. 57 for rabbit atrial cells), there would be insufficient current available to generate the largeV˙ max (167 V/s in Ref. 56) and nanoampere-range current amplitudes typical of the AP upstroke. In their model, Lindblad et al. (36) use the temperature correction suggested by Colatsky (10) (+3 mV/10°C) to shift the model curves to more positive potential. In the case of the human data reported by Feng et al. (19) and Sakakibara et al. (42), the resulting shift of ∼5–10 mV still leaves an inactivation half-maximal potential (V ½) well negative of the value observed in other mammalian cells and insufficient to generate the required fast inward current during the AP.
Recent human atrial data from Schneider et al. (46) display more realistic values for the V ½ of steady-state inactivation. The I-V curve of Schneider et al. resembles that measured by Sakakibara et al. (42), and both show a profile similar to the model I-V curve of Fig.3, although the data of Sakakibara et al. display a less abrupt activation of the current. The time constants of activation measured by Schneider et al. agree with those of the model; however, rapid inactivation time constants are on the order of 1 ms for a wide range of potentials (between −60 and +90 mV). The model displays fast inactivation time constants in this range for potentials positive to −40 mV, but at more negative potentials (between −80 and −40 mV), fast inactivation time constants increase in the model (to a maximum of 25 ms at about −65 mV). The rapid kinetics of fast inactivation over a wide voltage range reported by Schneider et al. may be due in part to their use of a fitting procedure allowing for overlap of activation and inactivation processes. It is unclear whether fast inactivation slows down in the study of Schneider et al. at potentials negative to −60 mV. They do not provide any fast time constants for recovery from inactivation at potentials between −95 and −60 mV, making it difficult to determine the value of τh at these potentials. On the basis of their recovery from inactivation data, there appear to be three time constants involved, whereas the data of Sakakibara et al. suggest two time constants. Given the difficulties in interpreting the data in these studies within the context of the simple formulation of Eq. 3, we chose to formulate the kinetics ofI Na exactly as in the work of Luo and Rudy (37), which nevertheless matches experimental data quite well, as shown in Fig. 2.
Transient outward current.
I to has been shown to differ substantially in its faster recovery kinetics in humans than in other mammalian species (20,47). On the basis of voltage-clamp current measurements, we found thatI to activation was best fit using a cubed activation gate. Activation and inactivation were fit to data from Wang et al. (55). A single inactivation gate was included in the model. In some current traces, slow inactivation can be seen, but this was attributed to slow inactivation of I Kur and was not included directly in our formulation of I to. The simulated current recordings shown in Fig. 6 indicate that the model reproduces the essential properties of experimental recordings ofI to + I Kur (55).
Ultrarapid delayed rectifier current or sustained outward current.
I Kur has been shown to play an important role in AP current balance at plateau potentials in human atrial myocytes (1, 47,55). On the basis of data from Wang et al. (55), we found thatI Kur had activation kinetics similar to those ofI to but activated at more negative potentials and displayed only partial slow inactivation. Those features are included in our formulation of the model. As with I to, changes in I Kur density can have considerable effects on AP morphology, as shown in Fig. 20.
L-type Ca2+ current.
I Ca,L has been studied extensively in human atrial tissue (9, 32, 35, 39, 49). Its complex inactivation kinetics, mediated in part by intracellular Ca2+, are only partially understood (49). This has made modeling the current more difficult and highly dependent on an accurate formulation of intracellular Ca2+ homeostasis and its control by the SR system (seeSR Ca2+ handling and buffering). Our formulation follows that of Luo and Rudy (37) and includes voltage-dependent activation and inactivation gates as well as a Ca2+-dependent inactivation gate. We have made modifications to the original formulation of Luo and Rudy on the basis of available experimental data (35, 49). However, as discussed by Luo and Rudy, fundamental questions regarding the complex voltage- and Ca2+-driven inactivation of I Ca,Lremain, from an experimental and a modeling standpoint. For example, data from Sun et al. (49) suggest that strictly voltage-dependent inactivation of I Ca,L is incomplete even for very long inactivation pulses. This suggests that steady-state inactivation ( f ∞) saturates at an inactivation fraction greater than zero (0.45 measured in Ref. 49). As reported by Luo and Rudy, we found that implementing such a relationship gave rise to uncharacteristic results, such as large I Ca,Lwindow currents observed on reuptake of Ca2+ during long APs. Another example can be found in the data of Sun et al., where an apparent second Ca2+-dependent inactivation time constant, on the order of 300–400 ms, could be observed during inactivation of I Ca,L. This is not accounted for in the model, and its origin has not been clarified experimentally. It may be that additional dynamical complexity in I Ca,Linactivation arises as a result of a more complex time course of the Ca2+ transient. This is in agreement with data from Hatem et al. (26) that point to rapid and slow phases during the rise of the Ca2+ transient in human atrial myocytes. These factors are related to limitations in our modeling of the SR Ca2+handling, which represents a simplified formulation of a temporally and spatially complex intracellular process (see SR Ca2+ handling and buffering andLimitations).
Delayed rectifier current.
I K has been characterized by Wang et al. (54) in human atrial cells. As in other mammalian species (44, 45),I K in human myocytes is the sum of a rapid and a slow component. We adopted the current formulation of Zeng et al. (61) fit to human atrial data (54). Steady-state activation and inactivation curves from human atrial cells are similar to those in other species (e.g., rabbit atria in Ref. 45). As pointed out by Zeng et al., there is conflicting evidence regarding the activation and deactivation kinetics of both currents. Wang et al. concluded that activation was well fit with a single time constant for both currents; however, they measured extremely slow activation time constants forI Ks. When included in the model, these slow kinetics limit the role of I Ks during the APO. Recent evidence from experiments on rabbit and guinea pig ventricular myocytes suggests fast and slow time constants for activation and deactivation of I Kr and I Ks(27, 43). Fast time constants for activation of I Ksreached a maximum of 330 ms at +10 mV and 37°C. We account for possible faster kinetics of I Ks in the model by using time constants that are one-half the values measured by Wang et al., putting the maximum time constant for I Ksactivation at ∼500 ms at +10 mV. By use of this formulation,I Ks plays a more significant role in repolarization during the AP but remains much less important thanI Kr at the lower plateau potentials typical of the human atrial AP.
SR Ca2+ handling and buffering.
SR Ca2+ handling and buffering involve uptake, storage, and release of Ca2+ from various cellular compartments (58). We followed the formulation of Luo and Rudy (37) but included a modification to the release mechanism suggested by Friedman et al. (21,22). Release of Ca2+ from the JSR compartment depends on the flux of Ca2+ through the sarcolemmal and SR release channels. This formulation aims to reflect the close coupling of L-type Ca2+ channels in the sarcolemma and JSR Ca2+release channel (34). For closely juxtaposed channels, one can expect that true local Ca2+ concentration in the vicinity of the release channel is more accurately reflected by local Ca2+flux than by bulk cellular Ca2+ concentration. The formulation is also adapted to afford a greater sensitivity of the release mechanism to flux through the L-type Ca2+ channel, as opposed to Ca2+ entry through the Na+/Ca2+ exchanger (22). Release is triggered when the computed Ca2+ flux exceeds a predefined threshold through the opening of an activation gate u. Inactivation is mediated through one flux-dependent (v) and one voltage-dependent (w) inactivation gate. An added advantage of this formulation is that it is completely analytic and does not require identification of qualitative trigger events such as the occurrence ofV˙ max.
As shown in Fig. 13, Ca2+ transients elicited in the model are qualitatively similar in amplitude and late time course to the experimental transients recorded in response to voltage-clamp steps to 0 mV (26). The discrepancy in the initial rise of the transient has potentially complex causes. The experimental transients were recorded at room temperature, which may in part explain their slower rise. In addition, their rising phase included rapid and slow components. It was postulated by Hatem et al. (26) that the slow rising phase was due to Ca2+ release from sites not closely juxtaposed to sarcolemmal L-type Ca2+ channels. Because of the absence of intracellular Ca2+ diffusion or specific myoplasmic Ca2+ compartments in our model, it is unrealistic to expect to be able to directly reproduce these features of the transient. We may consider the modeled transient as a compromise formulation that reproduces the rapid rise in local Ca2+ concentration expected in the vicinity of the sarcolemma. This ensures our ability to use [Ca2+]i as a direct inactivator ofI Ca,L through the f Ca gate. An updated formulation of intracellular Ca2+ handling should include local myoplasmic compartments in an attempt to capture the essential features of intracellular Ca2+ diffusion.
Inward rectifier K+ current.
I K1 plays a major role in the late repolarization phase of the AP and in determining resting membrane potential and resistance. As stated earlier, there is considerable variability in the measured input resistance of isolated human atrial cells (1, 52), with resting potentials often displaced to positive potentials compared with multicellular tissue preparations (1, 52, 54, 56). Our formulation ofI K1 was adjusted to reproduce the slow late repolarization rate observed in human atrial AP recordings. In doing so, we were able to produce a realistic resting membrane resistance of 180 MΩ (52) and a resting potential characteristic of healthy human atrial multicellular preparations of around −80 mV (56). The modelI-V relationship for I K1 (Fig. 4) has a shape characteristic of direct recordings in human atrial cells (31), but the specific parameters were adjusted to match the AP characteristics described above.
Behavior of the Model AP
Control model AP.
The control model AP exhibits a spike-and-dome morphology. As discussed below, several AP morphologies have been recorded from human atrial cells. In the model the transient outward current produces the initial phase 1 repolarization after the I Na-driven upstroke. On inactivation of I to,I Ca,L, which remains after rapid Ca2+-dependent inactivation, causes a slight depolarization of the membrane and produces the dome of the AP. This net inward current is eventually countered by the slowly activating outwardI K, resulting in slow repolarization. Slow late repolarization is a balance between deactivatingI K, time-independent I K1, and opposing inward current generated by the Na+/Ca2+ exchanger during the late phase of the AP.
Na+/Ca2+ exchanger.
The Na+/Ca2+ exchanger plays an important role in determining APD. As in experiments, we have shown that the exchanger is partly responsible for producing the slow late repolarization typical of human atrial APs. Block of the exchanger resulted in a significant reduction in APD90 (30% in Fig. 19). Because of the slow repolarization rate and small current amplitudes during late repolarization, APD90 is sensitive to small changes in late AP currents such as I K,I K1, and I NaCa. This may explain some of the variability in APD90 measurements from human atrial cells and tissues (5, 23, 24, 32, 33).
Rate dependence of model AP.
Rate dependence of the model AP is described in Fig. 16 and 17 and Table 3. The model is consistent with experimental data in several respects. First, it exhibits little rate dependence at frequencies <1 Hz, consistent with data from Fermini et al. (20) and Le Grand et al. (32). The model predicts greater relative AP shortening at APD50 than at APD90. This is seen in the data of Le Grand et al., even though their APD90 values are somewhat greater than those of the model. Comparing the top panels of Fig. 16, we observe that in experimental recordings and model simulations the AP waveforms at various rates tend to converge at some positive potential before complete recovery. Late repolarization is slowed at faster rates, in part because of a reduced activation ofI K in APs with shorter, more negative plateaus that opposes the rate-dependent increase in I Kavailability. This tends to reduce rate adaptation of the model AP measured using APD90 and is a direct consequence of the sensitivity of late AP repolarization to small changes inI K, I K1, andI NaCa, as discussed above.
Variability in AP morphology.
Variability in AP morphology is often observed in recordings of human atrial APs. Early observations of APs from isolated human atrial tissue demonstrated a considerable variability in their morphology (17, 24,50, 51). Some cells showed little phase 1 repolarization and a positive plateau; others exhibited a prominent phase 1 with a "spike-and-dome" morphology (17, 24, 50). It was suggested that age-related differences may be due to changes inI to (17). Recently, there have been attempts to define various types of APs on the basis of their morphological characteristics and/or underlying ionic current densities (1,4, 54). Wang et al. (54) identified three "types" of APs on the basis of morphology, ranging from type 1, a rectangular AP with a positive plateau, through type 2, a spike-and-dome AP with a plateau at ∼0 mV, to type 3, a triangular AP with little plateau. They associated these changes in morphology with a measured increase in the relative density of I to to I Kfrom type 1 to type 3 APs. Benardeau et al. (4) recently characterized two AP types in their isolated human atrial cell preparations: type A APs are spike-and-dome APs with a high plateau similar to Wang's type 1 or 2 APs; type B APs were triangular APs similar to Wang's type 3. In an attempt to identify the source of differences in AP morphology between types A and B, these investigators found no discernible difference in I Ca,L density between the two cell types. All three AP types have been recorded from human atrial multicellular preparations with fine-tipped microelectrodes (38).
The control model AP is typical of the type 1 APs described previously (38, 54). Our investigation of the role of I to,I Kur, and I Ca,L in AP morphology in Fig. 20 reveals that, in our model, variations inI to alone are able to generate many of the variations in AP morphology observed experimentally. AP shapes range from a triangular morphology (type 3 in Refs. 38 and 54 and type A in Ref. 4) for large I to, through a spike-and-dome morphology (type 1 in Refs. 38 and 54) at intermediateI to conductances, to a rectangular morphology (type 2 in Refs. 38 and 54 and type B in Ref. 4) for smallI to amplitudes. In contrast, variations inI Kur and I Ca,L alone cannot reproduce the rectangular morphology observed only in the presence of a small I to. The role of I to as an important determinant of AP morphology is supported by current measurements (54) showing that changes in AP morphology types (1, 2, and 3) were accompanied by changes in the relative amplitude ofI to consistent with the results of Fig. 20. An increase in I to density was also shown to accompany the change from rectangular to spike-and-dome AP morphology observed in young vs. adult human atrial myocytes (17). Although the amplitude ofI to appears to be an important determinant of AP morphology, it is clear that it cannot be considered independently of other membrane currents and that AP morphology will depend on the relative amplitudes of all currents involved.
Clinical Relevance
Atrial fibrillation (AF) is the most common sustained arrhythmia in clinical practice and that for which treatment remains the most problematic (38). The AP model provides insights into the basic mechanisms underlying a variety of important clinical determinants of AF. Impaired atrial APD accommodation to rate has been observed in patients with AF (5) and likely underlies the flat refractory period-heart rate relationship of patients susceptible to AF (2). Our model points to rate-dependent I Ca,L inactivation combined with incomplete I K deactivation as the basic mechanism of APD accommodation to rate. Work by Li and Nattel (35) suggests that inhibition of I Ca,L alone is sufficient to abolish rate adaptation of the human atrial AP, and it has been shown that human myocytes from dilated atria have a substantially reduced I Ca,L (32, 39). In the present study, strong I Ca,L reduction alone was sufficient to abolish rate adaptation of the model AP (Fig. 18). Although I K and I Ca,Lcontribute to model AP rate adaptation, important reductions inI Ca,L lower plateau height to the point whereI K is greatly reduced and AP accommodation is virtually abolished. The model explains how strong reductions inI Ca,L, as caused by atrial disease in humans or by exposure experimentally to I Ca,L blockers, can largely abolish AP rate adaptation and contribute to the susceptibility to AF.
Another property strongly associated with the occurrence of AF is AP heterogeneity (38). The present model indicates that variations in plateau currents (particularly I to) over the experimentally measured range can account for much of the AP variability reported in human atria. These mechanistic insights hold the promise of an understanding of how drugs that alter ionic currents can affect these important AP properties and, thus, may help in the development of improved therapeutic approaches to treating atrial reentrant arrhythmias like AF.
Potential Limitations
Although we have tried to develop a model that is closely based on experimental data, limitations exist because of availability of data and considerable variability in existing experimental data. Limitations of this work are similar to those discussed for recent ionic models (36). The following issues should be considered in evaluating the model.
Data from human atrial cells and tissues typically display considerable variability, with a wide range of current densities having direct consequences on AP shape, rate dependence, and response to pharmacological intervention. The use of averaged data from selected preparations does not take this into consideration and introduces possible complications in comparing model output with experimental AP recordings under various experimental conditions. This is in part why we have considered carefully the effect of varying specific current conductances on AP morphology. In addition, the temperature dependence of current kinetics and amplitude is often poorly characterized and must be approximated, with the help of available data, during the course of model development.
When possible, we have attempted to show experimental data along with corresponding simulated behavior. However, we note that, for voltage-clamp current recordings and experimental I-Vcurves, there may be significant bias in the measured current amplitudes. Selection factors such as the ability to record measurable currents and the viability of cells contribute to bias, especially with currents that are difficult to measure or display significant variability, such as I Ca,L orI K. These amplitude-dependent selection factors are unlikely to bias the measured kinetics of a given current. Because of bias in current amplitude measurements, it is rare that exact current densities reported in experimental studies can be incorporated directly into a model and produce a realistic AP morphology. When necessary, current densities from experimental data were adjusted in the model to improve overall AP properties. The same limitations apply to direct comparisons of current recordings to corresponding model output. Such comparisons can be misleading unless one selects a current trace with an amplitude that exactly matches the amplitude used in the model or unless a scaling factor is used to match the peak current amplitudes. This is why we chose to present scaled experimental data for current recordings and I-V curves, with the scaling factor selected to match the maximum current amplitudes in the experiments to those in the model. Whenever such scaling is carried out, we ensured that the corresponding scaling factor was explicitly given to allow comparison of current magnitudes between model and experimental data.
Limited data are available on intracellular Ca2+ dynamics, Ca2+ storage and release kinetics from the SR, and its role in the inactivation of I Ca,L. We have used a modified version of the intracellular Ca2+-handling scheme proposed by Luo and Rudy (37). This produces a realistic intracellular Ca2+ transient in response to suprathreshold Ca2+ flux into the cell. However, recent experimental results suggest that Ca2+-dependentI Ca,L inactivation may involve complex local phenomena requiring an additional level of model complexity (49), beyond an accurate representation of the average Ca2+transient. This is also important in view of the sensitive dependence of the model AP and its rate dependence on the dynamics of late repolarization, controlled in part by the Ca2+-dependentI NaCa.
Our formulation of the delayed rectifier current represents a compromise between a simple current representation and available experimental evidence that I Kr andI Ks may display complex, multiexponential deactivation kinetics (27, 43). Slow deactivation of these currents may have important effects on late repolarization, refractoriness, and AP rate dependence. Whenever possible, we have tried to select the simplest formulation that captures the essential features of available experimental data. It is possible that the complex kinetics of some currents, such as I K, require more detailed current formulations involving several channel states not readily modeled using a simple Hodgkin-Huxley formalism.
We have chosen not to consider time-varying extracellular ion concentrations in our model at this stage. Certain currents, such asI K1, are known to depend on extracellular ion concentrations. Accumulation of ions in the extracellular cleft space may be an important modulator of ionic currents and AP characteristics at fast pacing rates. These phenomena may be taken into account by modifying appropriate current formulations and including dynamic extracellular cleft space ion concentrations in the model.
Finally, our model was adjusted to produce stable ionic concentrations at rest. In response to periodic stimulation, ionic balance is disturbed and slow changes in intracellular ionic concentrations occur in the model. Initial transients due to kinetic rate adaptation of the currents dissipate over a few seconds. However, small changes in ionic concentrations can take several minutes to develop and result in slow changes in the morphology of the AP. These changes are dependent on the nature and the formulation of the stimulus. The long-term behavior of the model depends on whether the charge applied through the stimulus is carried by an ion, the balance of which is explicitly included in the model. The choice of the ion selected to carry the stimulus charge also affects the outcome. We have chosen to simulate the model without the stimulus having a direct effect on the ion concentrations. The sensitivity of long-term ionic concentrations to stimulus formulation and frequency is a property of ionic models that attempt to preserve a detailed ionic balance (25a, 50a). When discussing results under periodic pacing, we have chosen to present model results after initial rapid transients have dissipated, after 12 s of pacing from rest. The AP morphology at this point is independent of the form and implementation of the stimulus and takes into account the kinetic adaptation of the currents at the specified pacing frequency. However, it does not include the effect of concentration changes occurring on a slower time scale in the model. This is also true of the experimental data on which the model is based. Most frequency-dependent data are collected with the assumption that preparations stabilize within seconds of the onset of stimulation. A detailed investigation of the long-term dynamics of the model and its dependence on the formulation of the stimulus is left for further study.
The authors thank Drs. Jianlin Feng, Normand Leblanc, Gui-Rong Li, Hui Sun, and Zhiguo Wang for help with experimental data and Jean Perrault and Charles Dupont for the use of computer equipment.
Address for reprint requests: M. Courtemanche, Research Center, Montreal Heart Institute, 5000 E. Belanger St., Montreal, PQ, Canada H1T 1C8.
This work was funded by grants from the Medical Research Council of Canada (S. Nattel) and the Natural Sciences and Engineering Research Council (M. Courtemanche). M. Courtemanche is supported by a Chercheur-Boursier Scholarship from the Fonds de Recherche en Santé du Québec.
- Copyright © 1998 the American Physiological Society
Some fractional equations require evaluation of a limit to determine their values at membrane potentials for which their denominator is zero.
Differential Equations
Instantaneous equilibration of Ca2+ with buffers is assumed in all cases Equation 18 Equation 19 Equation 20 Equation 21 Equation 22 Equation 23 Equation 24 Equation 25 Equation 26 Equation 27
Equilibrium Potential
Equation 28
Fast Na+ Current
Equation 29 Equation 30 Equation 31
Equation 33 Equation 34
Time-Independent K+ Current
Equation 35
Transient Outward K+ Current
Equation 36 Equation 37 Equation 38 Equation 39 Equation 40
Ultrarapid Delayed Rectifier K+ Current
Equation 41 Equation 42 Equation 43 Equation 44 Equation 45 Equation 46
Rapid Delayed Outward Rectifier K+ Current
Equation 47 Equation 48 Equation 49
Slow Delayed Outward Rectifier K+ Current
Equation 50 Equation 51 Equation 52
L-Type Ca2+ Current
Equation 53 Equation 54 Equation 55 Equation 56
Na+-K+ Pump Current
Equation 57 Equation 58 Equation 59
Na+/Ca2+ Exchanger Current
Background Currents
Equation 61 Equation 62
Ca2+ Pump Current
Equation 63
Ca2+ Release Current From JSR
Equation 64 Equation 65 Equation 66 Equation 67 Equation 68
Transfer Current From NSR to JSR
Equation 69 Equation 70
Ca2+ Uptake Current by the NSR
Equation 71
Ca2+ Leak Current by the NSR
Equation 72
Ca2+ Buffers
Equation 73 Equation 74 Equation 75
Numerical Integration
At time step p, the updated value of a time-dependent variable is given by Equation 76for χ = V, any time-dependent ionic concentration, and
Equation 77
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After years of hating my nose I decided to get a nose job exactly 2 weeks ago. Do far so good and I am happy with the appearance of my new nose. However, I can feel some small scar tissue building right inside my nose. Does this go down over time or something that has to be removed?
Scar Tissue Inside the Nose Go Down After Rhinoplasty?
Doctor Answers 6
Scar Tissue After Rhinoplasty
Having some scar tissue after rhinoplasty surgery is typical, but as other surgeons here have mentioned it will improve over time. I find it seldom necessary to remove. However in the case that scar tissue causes some sort of obstruction then it must be excised.
Scar tissue two weeks after rhinoplasty
Since it has only been two weeks since your rhinoplasty, it is probably not scar tissue that is present. More likely is fluid retention inside the nose or a piece of swollen cartilage from your tip surgery. This should settle down over the next several months.
Learn more: http://www.seattlefacial.com
2 Weeks Too Early To Assess Scar Tissue in Rhinoplasty
The intial swelling with rhinoplasty can take a while to resolve. During that time you can sometimes feel some hard scar tissue building up. It typically resolves with time.
Do not try to feel your nose, instead look at it. If you SEE a swelling or asymmetry bring it to the attention of your surgeon. It can be resolved with Kenalog injections in early recovery period.
Learn more: http://www.facialplastics.net
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Scar tissue 2 weeks after rhinoplasty.
This is probably only swelling and will require nothing more than patience to resolve. Your surgeon will let you know in 6 months if this has not gone away.
Scar Tissue, Swelling Inside of Nose After Rhinoplasty
Hi Huskies,
Glad to hear that you like your new nose. Your nose will take months to fully heal, and swelling will progressively decrease. You should keep your fingers and anything else smaller than your elbow out of the inside of your nose after rhinoplasty. If you have the need to feel something inside of your nose, see you rhinoplasty surgeon and let him/her examine you. Good luck and be well.
Dr. P
Swelling after rhinoplasty
You should be experiencing a lot of swelling at this point, and the scar tissue and swelling inside the nose will go down with time. Be patient, as this can take many months, and sometime up to a year, to fully resolve. You may also notice that some parts of the nose will remain swollen longer than others, and this is not unusual. Best wishes, /nsn.
These answers are for educational purposes and should not be relied upon as a substitute for medical advice you may receive from your physician. If you have a medical emergency, please call 911. These answers do not constitute or initiate a patient/doctor relationship.


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burn (one's) bridges
1. Literally, to destroy a bridge or path behind oneself, so that others cannot follow. This usage is often related to military action. When the troops retreated from the area, they were sure to burn their bridges behind them.
2. To do something that cannot be easily undone or reversed in the future (often because one has behaved offensively or unfavorably). I think you really burned your bridges when you announced you were quitting and proceeded to insult your boss in front of the whole staff. She's young, so I don't think she realizes that she'll be burning her bridges if she goes to work for their competitor.


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Lincoln, Neb.-- After suffering a traumatic brain injury, 24-year-old Isra Somanas is getting back on his feet and dancing.
Somanas was hit by a car while riding his bike at the intersection of 40th and O Streets on July 15.
After being in a coma for several days, doctors decided the next step of recovery for Somanas would be rehabilitation at Madonna Rehabilitation Hospital.
Somanas went there in mid-July, not being able to walk or talk. Now nearly six months later, he's flying around a dance floor.
Somanas' therapist says it has been a remarkable recovery because he's made so much progress in such a short amount of time.
Although riding a bike is his passion, Somanas says he is thankful he has the ability to dance.
"One of the things I thank God for continuously is that I can still dance. I can head out to Pla Mor and hit the dance floor and just dance the night away. I just really love dancing. I love moving to music and things like that," Somanas said.
Doctors say Isra Somanas can ride a bike again one year after the accident which will be July 15. Somanas says he does plan to do that.
He is also finishing the semester at UNL and plans to graduate with a degree in mechanical engineering.
Somanas moved from Bangkok, Thailand four years ago to study at UNL. His family did make the trip to the U.S. after hearing about the accident.


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Driving sales with an established brand at multiple retail accounts is hard enough for the biggest consumer packaged goods companies. Creating a new brand in a nonexistent category is even more daunting as a pair of entrepreneurial family members discovered a decade ago when they launched the Granite Gold line of stone-care products. This is their story.
Two stone guys with no experience in consumer package goods or retail set out to conquer the world by trying to fit a square peg into a round hole.
After spending a few years in the late 90s developing a consumer DIY solution for proper stone care — literally creating and mixing formulas, and testing them on granite, marble, travertine and all other natural stone because of variances in porosity and density — we figured we had a strong case to bring Granite Gold to a mass audience.
We quickly encountered our first lesson. There wasn't a category specific for stone-care products and they didn't necessarily fit neatly into any of the existing categories. We were able to pique the curiosities of retail buying executives who, in many cases, had natural stone in their own homes — and our deep experience in the stone industry helped with our credibility. Yet, the results of those meetings were the same time and again: Where does this fit in the store? Do consumers need it or want it? How will we get the consumer into the store to buy Granite Gold?
Those doses of humility and reality set the course in our formative years as we developed a strategy to respond to those challenges. It started with "get the consumer first."
From 2003 to early 2005 when the building boom was still growing, we hit the road with a product sampling program for local home builders and distributors in the San Diego area. We performed what seemed like an infinite number of how-to demonstrations at regional home and garden shows and in Costco club stores. This led to Granite Gold becoming the no. 1 Costco road show in San Diego and the first of many rotations across the U.S. While that endeavor proved effective, we discovered that homeowners had little to no knowledge about how to care for their natural stone properly — particularly the damage common household cleaners do to natural stone surfaces.
Product efficacy, our expertise in stone care and our willingness to take the time to teach homeowners the essentials of stone-care maintenance became the foundation upon which we built a loyal consumer base. At the time, Granite Gold wasn't readily available at retail stores, but consumers were buying our products in droves from our website thanks to word of mouth.
With our consumer base limited to Southern California, it still wasn't enough to convince retail buying executives to get on board. So in 2005 we set out to expand our footprint and create demand for Granite Gold. We launched a national consumer awareness campaign that year on the dangers common household cleaners pose to natural stone surfaces. That effort proved effective because it expanded our footprint, got consumers to ask local stores to carry Granite Gold and got retail buyers and brokers to meet with us.
After Costco, Raley's, Bel Air and Nob Hill gave us our first shot in grocery, Albertsons, SuperValu, Fred Meyer and Smart & Final and Unified Western Grocers soon followed. Delivering on our promise to hold in-store demonstrations on proper stone care and how to use Granite Gold helped immensely and has proved to be our most effective sales tools.
As we entered the retail marketplace, we used every communication tool at our disposal — email, Web, postal mail, direct-mail coupons — to drive consumers to stores to buy Granite Gold. Frankly, it was more cost-effective for consumers to buy products at retail stores instead of via our website because of shipping and handling.
Realistically, we were only scratching the surface. We understood we didn't have the knowledge or experience necessary to build a brand, grow distribution and create a category at retail and mass. That's when we brought in a CPG sales and marketing veteran from WD-40 Co. who knows a thing or two about supply chain, packaging, retail, mass, big box, and so on. We went into a sort of bunker mode with Scott Martin, formerly SVP at WD-40 Co., and developed a national sales and marketing strategy. Moreover, we found a solution to our greatest challenge at the time: supply chain.
We were able to handle the boon in our online sales with relative ease, but as we began to distribute more to the retail environment, we realized we were in over our heads in meeting demand. Fortunately, we discovered contract manufacturer Goodwin Company, a long-trusted and valued partner to this day.
With supply chain management solidified, we expanded distribution in 2006 to Orchard Hardware Supply and True Value hardware stores and Ralph's and Bristol Farms grocery stores. That was our first of several banner years, doubling business growth at year-end, and shortly after, in 2007, we achieved another milestone: Bed Bath & Beyond, our first national account.
Throughout the next half-dozen years, we doubled our product portfolio from the five SKUs we started with and expanded distribution in grocery, hardware, mass merchant and specialty store channels throughout the U.S. and internationally.
It really just started out as an idea to create a product to help our installation and restoration businesses' customers. Little did we know back then we were building what would become a widely recognized brand known for great stone-care products. We're still relatively young with more room to grow and plenty more to learn, and the one constant will remain: sharing our expertise about stone care and using that expertise in product development. It's old-school customer service, and it works.
Lenny Sciarrino is president, CEO and co-founder of Granite Gold Inc. He started the family company 10 years ago with his cousin, Lenny Pellegrino, who still runs his own restoration business in Southern California. Granite Gold brand stone-care products are available in grocery, hardware, mass merchant and specialty store channels throughout the U.S. and internationally.


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STAND-IN captain Shane Watson has warned that Australia must rekindle its killer instinct, describing its performance in the drawn five-match ODI series against the West Indies as very disappointing.
The tourists claimed a 30-run win in Sunday's final match in St Lucia, levelling the series 2-2 and thereby avoiding a first ODI campaign defeat to the West Indies in 17 years.
But, as in previous games of the tour, Australia allowed the Windies back into the match just when an easy victory seemed assured.
Chasing 282, the hosts' challenge appeared finished when captain Darren Sammy strode to the crease with the score 7-118.
What followed was the innings of Sammy's career, smashing 84 off 50 balls as he and Andre Russell (41) scored 101 off just 59 deliveries for the eighth wicket to drag the Windies back into the match.
It reflected similar moments in the series where Australia took its foot off, such as the tied third game in St Vincent where the hosts recovered from 6-117 to all but snatch a win.
This time Australia had enough runs on the board to repel Sammy's blitz and claim victory, but Watson said a ruthless approach was required to overcome the worrying trend.
''We're at a stage where things look like we're cruising OK and change quite quickly,'' Watson said.
''That's happened since I've come back in the Australian team over the last 10 games. It's something we certainly need to work on.
''To tie the series, especially when in a number of those games we were at a stage where we should have buried the West Indies, it's probably very disappointing.
''We have to get better at those stages because that's where we are really going to grow as a team.''
Watson hands over the captaincy to George Bailey for the two-match Twenty20 campaign starting in St Lucia tomorrow morning (Melbourne time), and he expects it to be just as competitive as the ODIs.
''They've got a very good Twenty20 team as we do,'' Watson said. ''I think it's just going to continue to be a great series.''
Sammy too was ruing missed opportunities after his team just failed to secure the Windies' first ODI series win over Australia since 1995.
The all-rounder was the man run out when Australia snatched the tie in the third match of the series, and he admitted that moment would probably haunt him.
''Every time I close my eyes, this series will reflect in my head and no doubt St Vincent will definitely come to play,'' he said.
And a heavy collision between Matthew Wade and Ben Hilfenhaus during the game brought back horrible memories for Australian cricket fans.
Wade, sprinting back from his wicketkeeping position, barrelled into Hilfenhaus' knee as the fast bowler claimed a high catch during the win.
There was immediate concern as Wade lay on the ground in agony following the collision and thoughts went back to the collision between Steve Waugh and Jason Gillespie in Sri Lanka in 1999, which left Gillespie with a broken leg.
Thankfully for Australia this time the worst injury sustained was Wade being winded.
S WATSON c Samuels b Russell 66
D WARNER c D Bravo b Narine 69
P FORREST c Roach b Russell 53
G BAILEY c Russell b Roach 19
B LEE b Narine 12
M HUSSEY c Pollard b Russell 25
D HUSSEY c Baugh b Russell 0
M WADE c D.M. Bravo b Roach 26
C McKAY c D.M. Bravo b Roach 0
X DOHERTY not out 1
B HILFENHAUS not out 0
Sundries (2b, 1lb, 7w) 10
Total (9 wkts - 50 overs) 281
Fall of wickets: 118 (Warner), 161 (Watson), 193 (Bailey), 212 (Lee), 241 (Forrest), 241 (D Hussey), 279 (M Hussey), 279 (McKay), 279 (Wade)
Bowling: K Roach 9-0-53-3 (1w), D Bravo 6-0-40-0 (3w), S Narine 10-0-55-2, D Sammy 6-0-29-0 (1w), A Russell 9-0-61-4 (1w), M Samuels 10-0-40-0 (1w)
West Indies
J CHARLES c and b Lee 0
A BARATH c Bailey b Doherty 42
M SAMUELS c Wade b Lee 0
D.M. BRAVO c Wade b McKay 3
D BRAVO c Bailey b Watson 19
K POLLARD c M Hussey b Watson 33
C BAUGH c Hilfenhaus b Lee 13
A RUSSELL lbw b Doherty 41
D SAMMY c M Hussey b Hilfenhaus 84
S NARINE c Wade b McKay 7
K ROACH not out 2
Sundries (1lb, 6w) 7
Total (all out - 47.2 overs) 251
Fall of wickets: 0 (Charles), 5 (Samuels), 37 (DM Bravo), 63 (D Bravo), 76 (Barath), 111 (Baugh), 118 (Pollard), 219 (Russell), 232 (Narine), 251 (Sammy)
Bowling: B Lee 9-3-42-3, B Hilfenhaus 8.2-1-36-1 (2w), C McKay 10-0-68-2, S Watson 10-0-44-2 (3w), X Doherty 10-0-60-2 (1w)
Result: Australia won by 30 runs
Series: Five-match series drawn 2-2, with one tie.


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- October 3, 2007, 4:57 AM
NASA's longest serving administrator, Dan Goldin, will resign from the space agency November 17 after being in that position for 10 years. He accepted an interim position as senior fellow for the Washington-based Council on Competitiveness, an organization that works to establish U.S. economic competitiveness and leadership in world markets. It's too soon to know if Goldin's departure will have any effect on NASA's general aviation and other aeronautical research projects. He will work with NASA to identify an interim acting administrator before he leaves office


text:
HUDSON — Chrissy Baumaister silently pretended to pull the safety restraint over her shoulders and waved her arms over her head on an imaginary roller coaster ride.
Second grader Connor McCathron caught all the charades on a blue video-capable iPod nano — his first time using the tiny technology.
"Mrs. Baumaister, you were great!" Connor exclaimed to the Northwest Elementary School technology specialist after guessing what she was doing. "Oh my God! I'm recording you!"
His enthusiasm was contagious in the school's new Collaboration Cafe, which Baumaister organized to teach students and teachers ways to incorporate technology, art, music and other hands-on activities into their literacy lessons. At other centers in the room, children figured out animal riddles on a Smart board, took measurements of pumpkins and used a website to create posters of words they used to describe pumpkins.
Doyle Surrittee declared the session a good time.
"You get to do different stuff," he said, smiling. "That's how you learn things."
His teacher, Clarice Saparito, agreed this approach gets everybody involved.
"This is more fun to them," Saparito said. "They think they're playing, and they're really learning."
For too long, assistant principal Holly Oakes said, the emphasis in schools has been on compliance with rules and standards. Teachers felt the pressure, which in turn trickled down to students.
At Northwest, where four in five students qualifies for free or reduced-price meals, the faculty wanted to change that direction and focus on inspiring children to learn.
"The big thing that came out over and over again was student engagement," Oakes said.
That meant making the lessons more active, creative and fun. Teachers had to be free to try new things, infusing a variety of disciplines and styles into their classrooms.
"If we come in the room and see kids wrist-deep in finger paint, we're good with that. If they're skipping around the room, we're good with that," Oakes said, adding that the administrators have started giving teachers special pins and flair for using innovative teaching techniques.
One of the potential pitfalls, though, was some teachers' lack of training.
And Baumaister found that offering 30 minutes of support before school begins didn't cut it. So she proposed, then created, the Collaboration Cafe, a classroom with all the "toys" that kids love to use and that teachers can make part of their lessons.
She put it in what had been a computer lab, placing all the computers into classrooms "where they should be."
The school chose a literacy theme — predictions for October, visualization for November — and then Baumaister crafted lesson plans using the different forms of technology for teachers to learn to adapt for their classes. As classes move through the cafe monthly, the teachers learn how to incorporate iPods, Smart boards and the like in a hands-on environment with extra support and students on-hand to see in real time what works.
So far the reaction has been almost completely positive.
"I am having too much fun," teacher Tammy Hickey said as she learned about Smart boards with a small group of second graders. "It's nice to be able to apply it, take it back to the classroom and apply it again."
Children have written letters thanking Baumaister for letting them use iPods to make movies, too.
"I want them to be happy that they're coming in here," she said. "They're so excited. That's what we want."
Over time, the plan calls for teachers and students to build upon the skills they're learning in the Collaboration Cafe as they create their curriculum lesson plans. Simple videotaping will lead to making movies on the computer. Basic Smart board operation can give way to interactive reports.
That's all great to second grader Giulia Desiderio, who said she likes variety in the classroom as she worked with Hickey and some classmates to create a poster.
"If you do the same thing over and over again," she said, "it gets boring."
Which is exactly what Northwest Elementary is aiming to avoid.
"We're trying to make it so their brains are getting the stimulation they need to open up and stretch," Oakes said.
Jeffrey S. Solochek can be reached at email@example.com or (813) 909-4614. For more education news, visit the Gradebook at tampabay.com/blogs/gradebook.


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DeWalt DWHT33373LWW 25' Tape Measure Wounded Warrior Project #DWHT33373LWW
DeWalt DWHT33373LWW 25' Tape Measure Wounded Warrior Project #DWHT33373LWW
The DWHT33373LWW includes:
- 1x DeWalt DWHT33373LWW DWHT33373L Wounded Warrior Project Edition 25' Tape Measure
- Special Edition Wounded Warrior Project Tape Measure from DeWalt
- A full 10' of blade standout makes it even easier to take accurate, single person measurements
- Durable heat treated spring offers reliable blade retraction for years to come
- Duo-Durometer blade lock system offers increased comfort and durability
- Extra wide blade with high visibility markings makes it fast and easy to get the measurements you need
- End hook grabs the material from either the top or bottom for added convenience
- Integrated belt clip allows you to keep the tape measure handy at all times
The DEWALT 25 ft. Tape Measure is durably made with an anti-rust polymer blade and heat-treated spring and boasts 10 ft. of standout for easy measurements. The blade is 1-1/8 in. wide and has 1/8 in. markings for readability, and a handy belt clip keeps it within reach. The hook grabs surfaces from either top or bottom, allowing easy operation measure for measure.
Tech Specs:
- Brand: DeWalt
- Model: DWHT33373L
- Type: Tape Measure
- Hand Tool Type: Tape Measure
- Belt Clip: Yes
- Electronic: No
- Housing Material: Plastic
- Lockable: Yes
- Magnetic Tip: No
- Metric/SAE/USS: SAE
- Tape Length: 25'
- Weight: 1.1lbs
- Package Qty.: 1
Overall Rating for DeWalt DWHT33373LWW 25' Tape Measure Wounded Warrior Project #DWHT33373LWW
Very fast shipping. Product is better in person and supports a great cause.
Excellent product!!!!!
By stephen from Baltimore, MD on March 9, 2015
I would recommend this item to a friend.
great product, great service - can't ask for anything more
WWP tape measure
By Paul from Mansfield MA on January 15, 2015
I would recommend this item to a friend.
Check out the rest of our...
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text:
Didactic & Hands-on Workshop
Henry Schein Halas Parkville, VIC
Presented by Dr Thomas Giblin & Dr Lincoln Harris
3 Module Program | 42 CPD Hours
Module One Implant Foundation 17-18 February 2017
Module Two Aesthetic Prosthetic 24-25 March 2017
Module Three Surgical Success 28-29 April 2017
Note: While registering, you have the option of selecting all THREE MODULES or INDIVIDUAL MODULES
Venue: Henry Schein Halas
Level 9, 369 Royal Parade Parkville VIC 3052
Implant Dentistry has become common place in dental practice and many patients expect their dentist to offer some type of implant solution.
The purpose of this course is to give attendees a foundation of clinical knowledge and techniques in planning and carrying out implant dentistry. Most dentists have studied a lot of theory about implants, yet often lack the practical teaching resulting in common clinical mistakes, damaging the clinician's confidence.
In this course, training will occur through didactic and practical exercises that help you obtain greater knowledge regarding successful implant treatment and how to achieve clinical excellence!
Learning Objectives
-Effective treatment planning using models, risk analysis, radiography and photography
-Patient prepping, risk and consent
-Room setup & needs
-Clinical Photography
-Atraumatic extractions
-Socket/ridge preservation
-Surgical guides
-Single implants & bridged Implants
-Immediate placement in extraction sockets
-Straight & angled Implants
-Soft tissue considerations & flap design
-Suturing techniques
-Hard and soft tissue augmentation
-The use of CBCT
-Implant prosthetics, single & multiple, fixture & abutment level, screw vs cement retained restorations
-Chair side temporaries
-Wound management & post op medication and follow up
-Implant failure management
Events Administration
Cvent Online Event Registration Software | Copyright © 2000-2017 Cvent, Inc. All rights reserved.


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The Upper Echelon - A group dedicated to players who have achieved Grade 4 Rank in Halo 3.:| is not amused. Go away or else he will evolve to >:|Rawr I'ma Triton Rawr!!!
It would've been more funny if the red guy betrayed his own hog.
wow this is pretty funny
I dont need one.
hey like infection games and maps or like to make them (infection games and maps)then the infection is right for youHeres how you do itgo to my profile select groups then select 'the infection'finally select join group and send to all people
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text:
Research in Motion's report Thursday that it has reduced its losses and increased its revenue appeared to convince investors that the Canadian company's plans for the future were sound.
The results weren't spectacular by any means, though they did exceed analyst expectations of $2.5 billion in revenue. RIM reported a loss of $235 million on revenue of $2.9 billion. It's significantly less than during the same period last year, when the company reported $4.2 billion in revenue. But the fact that RIM reduced its losses — last quarter they reported a loss of $518 million on revenue of $2.3 billion — was a mark in its favor.
Shares of the BlackBerry maker shot up after the market's close when the company announced its earnings report, after closing up 2 percent in the day's trading. In the minutes after RIM disclosed its latest financial results, the stock shot up 17 percent and well over $8 per share. RIM shares had hit a nine-year low this week on news of an outage in Europe and Africa.
The news comes as the company implores investors to focus on the next quarter, when the company is planning to launch a new operating system in an effort to wow its core users and make modest gains in its subscriber base.
Combined with the company's strong presentation at a developers conference in San Jose, Calif., this week, the results appear to have encouraged investors to stay with the company.
Chief Executive Thorsten Heins said at that conference that the company is in a good position to come in third in the war for operating system market share, behind Google's Android and Apple's iOS.
To take that position, the company is pushing out its new operating system, BlackBerry 10, in January and promises devices will follow.
Since shifting its focus to software development, the company has all but halted rolling out new devices — a decision that showed in the earnings results.
Phone shipments dropped to 7.4 million from 7.8 million in the last quarter. Sales of the BlackBerry Playbook dropped to 130,000 from 260,000.
But with ambitious plans to roll out BB10 devices on six continents next year, the company is hoping to recoup some of those losses where it's the strongest — with global enterprise businesses and markets outside of North America.
Related stories:
Research in Motion shares rise after system preview
Research in Motion restores BlackBerry service after outage


text:
Nikhil Kothari, who's one of the brains behind many things ASP.NET, has just released an update to a cool utility he calls the WebDevelopmentHelper. WebDevelopmentHelper allows you to inspect the HTTP requests being sent to and from a web application, which can be very helpful in troubleshooting when something's going wrong.
Nikhil's now added support for viewing the traffic generated by requests made in the context of an AJAX UpdatePanel control, as well as the ability to inspect JSON data. He's also added some documentation for the tool, which should make it even easier to use.
So if you've wondered how to see what's being generated when you click a link or control that's inside an UpdatePanel...wonder no more. Download the WebDevelopmentHelper, and find out!
[Hat Tip: Scott Guthrie]


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Choosing a close up filter set?
Looking to add a close up filter set to my SAL24-70. On eBay prices are all over the place from very low to very high. Since these are 1 element optics is there all much difference in glass quality between the lower and higher priced models? Any recommendations appreciated.
Spend a little more and get something worth having. My dad taught me that cheap tools were always the most expensive because they either didn't do the job or needed ot be replaced often.
If you want to start, get the Raynox 150 and learn how to use it. For more magnification, try the 250 but, the 150 is fne on most moderate teles.
My photo blog: http://birdsnbugs.wordpress.com
My camera club porfolio: http://www.pacameraclub.com/bgrant.htm
RF Stock Portfolio - http://www.dreamstime.com/resp129611
EXIF is embedded in photos WSSA #51 as bg5700
None of them is good, but still there are huge differences in quality, unfortunately largely uncorrelated with the price, and no one tests them seriously. You may happen to find test images for some of them of you google hard enough. Mind some are post-processed, for example lateral CA are often removed effectively; Nikon cameras do it on-the-fly.
You may not need a
because the weaker ones won't magnify much with such a short lens. You will likely need at least +3 diopter to get anywhere; these tend to be thick and heavy. Smaller-diameter diopters on step-down rings work with some lenses but vignette with others so you must check.
Some better "achromat" options:
Canon 250D (+4)
Raynox DCR-250 (+8)
Marumi DHG achromat +5 or +3 (large diameters available)
Automatic extension tubes are a good alternative with short lenses, typically less degradation of quality, but more of a pain to fiddle with.
none. The rule is never buy anything with one element unless you know it's good.
Canon make/made one that worked quite well from images i've seen, and the raynox 150/250/202(raynox have 2 elements i think), etc are probably up there at the top of the macro filters list too.
if you have a flash and/or only do macro on sunny days then ET's may be the better option for a Zeiss lens. If you know what its max magnification is then you can find out how much mgnification you'll get with ETs
osu wrote:
Looking to add a close up filter set to my SAL24-70. On eBay prices are all over the place from very low to very high. Since these are 1 element optics is there all much difference in glass quality between the lower and higher priced models? Any recommendations appreciated.
I second getting a DCR250, but not the original canon filters...too pricey.
Alternatively, a set of Kenko extension tubes is good too. On a 50mm F1.8, you get get slightly more than 1x magnification.
Macro Photography Blog: http://orionmystery.blogspot.com
As some of the others have said, it is best to avoid the single element close-up filters. They can be sharpish in the centre, but most have a lot of fall off from the centre and there tends to be a lot of CA. Multi-element acromats are best. Although having said that it depends on what you want. If it is just a few creative shots of flowers, or something else where it's all central detail it might not be too much of a problem.
However, I'd question whether a close-up filter is your best bet with a 24-70mm zoom. The longer the focal length, the greater the magnification a given strength of close-up has. You would need a very powerful close-up filter to get much mangification even at the 70mm end. The problem is you will probably also need a larger diameter close-up lens than the powerful Raynox close-up lenses to avoid vignetting. Extension tubes may be a better bet.
If possible I would see if you could borrow some close-up lenses or try some out in a shop. That is to check whether the magnification is enough, and whether they cause vignetting. Vignetting is much less of a problem as the focal length increases.
OSU, I recommend just about anything other than 1-element close-up lenses - honestly you've better off cropping with most of them.
See http://johnsankey.ca/tripod.html#E for the close-up methods I use. A reversing ring costs less than most close-up lenses and using the jig I describe you'll get shots as good as your lens can deliver.
Some great advice here. I actually have the AF extension tube set for Minolta/Sony. It's the 3 section set. I just found it to be be too cumbersome to carry around and it's sat in my old film case for years. What is clear is that the macro filters are not a good option.
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