**10. Monitoring: Noninvasive determination of aortic pressure, aortic modulus (stiffness) and peripheral resistance)**

The aortic blood pressure waveform contains a lot of information on how the LV contraction couples with the aortic compliance and peripheral resistance. Since accurate measurement of aortic blood pressure waveform requires catheterization of the aorta, we have developed a noninvasive method to determine the aortic pressure profile along with the aortic volumeelasticity and peripheral resistance. Fig 28 displays such a constructed aortic pressure profile.

The input to the model consists of auscultatory cuff diastolic and systolic pressures, along with the MRI (or echocardiographically) measured ejection volume-time profile (or volume input into the aorta). The governing differential equation for pressure response to LV outflow rate *I(t)* into the aorta is given by [13]:

$$\frac{dP}{dt} + \lambda P = m\_a \left[ I(t) + T\_a \frac{dI}{dt} \right] \tag{1}$$

where (i) *ma* is the aortic volume elasticity *(dP/dV*), (ii) *Rp* is the resistance to aortic flow *(=P/Q),* (iii) λ*=ma/Rp* , (iv) *I(t)* is the monitored inflow rate into the aorta, and (v) *Ta* is the flow-acceleration period.

This governing equation is solved for measured *I(t)* and *dI/dt* during the systolic phase from time *T1* to *T3* (Fig. 28). For the diastolic-phase solution from time *T3* to *T4*, the right-hand side is zero. The solutions for diastolic and systolic phases are given below by equations (2) and (3) respectively.

### **Solutionequations:**

38 Biomedical Science, Engineering and Technology

R 9 43 cmH 0 s /L a 2 =− ⋅ Now, that we have determined the expressions for *Ra* and *Ca*, the next step is to develop an

**Formulating a Lung Ventilatory Index (LVI) incorporating Ra and Ca:** We now formulate

*R P RF a k LVI Ca*

Let us now obtain order-of-magnitude values of *LVI,* for a mechanically ventilated COPD

=

TV 0.5 <sup>20</sup> <sup>2</sup>

= =

( ) (TV)

<sup>1</sup> [15 / ][0.33 ][20 ] 2 2 ( COPD) [0.035 / ][0.5 ] <sup>2</sup>

<sup>1</sup> 15 / 0.035 / 0.33 2 2

− == =

Now, let us obtain an order-of-magnitude of *LVI* (by using representative computed values of *Ra, Ca, RF,* TV*,* and*Pk*) as abovefor a COPD patient with improving lung-status just before

<sup>1</sup> [10 / ][0.33 ][12 ] 2 2 ( COPD) [0.05 / ][0.35 ] <sup>2</sup>

<sup>1</sup> 10 / 0.050 / 0.33 2 2

− == =

Hence, for *LVI* to reflect lung-status improvement in a mechanically ventilated COPD patient in acute respiratory failure, it has to decrease to the range of *LVI* for an outpatient

In fig. 27, it is shown that for the 6 successfully-discontinued cases, the *LVI* was (2900) ± (567) (*cmH*2*O/L*)3; for the 7 failed-discontinuation cases the *LVI* was (11400) ± (1433) (*cmH*2*O/L*)3. It can be also observed that *LVI* enables clear separation between failed and

*R cmH Os L C s L cmH O RF a a*

*<sup>L</sup> <sup>P</sup> cmH O <sup>k</sup>*

*R cmH Os L C s L cmH O RF a a*

*<sup>L</sup> <sup>P</sup> cmH O <sup>k</sup>*

*LVI Outpatient L cmH O L*

=

TV 0.35 <sup>12</sup> <sup>2</sup>

= =

successful discontinuation, which again points to the efficacy of *LVI.* 

integrated index incorporating these parameters.

patient in acute respiratory failure:

=

successful discontinuation.

=

wherein

wherein

a Lung Ventilatory Index (LVI), incorporating Ra and Ca, as:

<sup>3</sup> 5654 ( / ) <sup>2</sup>

<sup>3</sup> 2263( / ) <sup>2</sup>

COPD patient at the time of discontinuation.

*cmH O L*

*cmH O L*

*LVI Intubated*

C 0.020 0.080 L / cmH 0 a 2 = − (16)

= (17)

*cmH Os L s cmH O*

*L cmH O L*

*cmH Os L s cmH O*

−

−

• During diastolic phase,

$$P\_d(t) = P\_l e^{\lambda(T-t)} \tag{2}$$

At *T = T4*, *Pd (T)*=auscultatory *Pad =P1* 

Fig. 28. Model computed cyclic Aortic pressure profile, compared with measured pressure values. The systolic phase from *T1* to *T3* is when blood is ejected into the aorta. The diastolic phase is from *T3* to *T4* [13].

• During systolic phase,

$$P\_s(t) = (P\_1 - A\_1)e^{-\lambda t} + e^{-bt} \left[ A\_1 \cos(\alpha t) + A\_2 \sin(\alpha t) \right] \tag{3}$$

where

$$A\_1 = \frac{m\_a a \cot(T\_a \mathcal{A} - \mathcal{I})}{\left(b - \mathcal{A}\right)^2 + a\phi^2}$$

$$A\_2 = \frac{m\_a a \left[\left(a\phi^2 + b^2\right)T\_a + \mathcal{A} - b - T\_a b\mathcal{A}\right]}{\left(b - \mathcal{A}\right)^2 + a\phi^2}$$

Also, as noted in Fig 28, the boundary values are: At t =T2, dPs/dt =0; at t =T2, Ps(T2) = auscultatory Pas= P2 ; at t = T3, Ps(t =T3) = Pd(t =T3) Hence, based on these boundary values, the following equations are to be solved.

$$\frac{dP\_s(t)}{dt}(t=T\_2) = -C\_1\lambda e^{-\lambda T\_2} + e^{-bT\_2} \left[ A\_3 \cos(\omega \theta T\_2) - A\_4 \sin(\omega \theta T\_2) \right] = 0\tag{4}$$

• At *t =T2, Ps =P2*. Hence, from equations (3 & 4), we get:

$$P\_s(t = T\_2) = P\_2 = C\_1 e^{-\lambda T\_2} + e^{-bT\_2} \left[ A\_1 \cos(\omega \theta T\_2) + A\_2 \sin(\omega \theta T\_2) \right] \tag{5}$$

• At *t =T3, Ps(t =T3) = Pd(t =T3).* Hence, from equations (2 and 3), we obtain:

$$P\_1 e^{\lambda(T - T\_3)} = (P\_1 - A\_1)e^{-\bar{\lambda}T\_3} + e^{-bT\_3} \left[ A\_1 \cos(a\sigma T\_3) + A\_2 \sin(a\sigma T\_3) \right] \tag{6}$$

We now solve equations (4, 5 & 6), to determine the unknown parameters *ma, Rp* (and *T2*).

Fig. 28. Model computed cyclic Aortic pressure profile, compared with measured pressure values. The systolic phase from *T1* to *T3* is when blood is ejected into the aorta. The diastolic

() ( ) 1 1 [ <sup>1</sup> cos( ) sin( ) <sup>2</sup> ] *t bt Ps t P Ae e A t A t*

− − =− + +

<sup>1</sup> <sup>2</sup> <sup>2</sup> ( )

ω

> λ

λ

− + <sup>+</sup> <sup>+</sup> <sup>−</sup> <sup>−</sup> <sup>=</sup> *<sup>b</sup> <sup>m</sup> <sup>a</sup> <sup>b</sup> <sup>T</sup> <sup>b</sup> <sup>T</sup> <sup>b</sup> <sup>A</sup> <sup>a</sup> <sup>a</sup> <sup>a</sup>*

− + <sup>−</sup> <sup>=</sup> *<sup>b</sup> <sup>m</sup> <sup>a</sup> <sup>T</sup> <sup>A</sup> <sup>a</sup> <sup>a</sup>*

λ

At t =T2, dPs/dt =0; at t =T2, Ps(T2) = auscultatory Pas= P2 ; at t = T3, Ps(t =T3) = Pd(t =T3) Hence, based on these boundary values, the following equations are to be solved.

2 2

2 2

2 2 2 21 1 22 2 ( ) cos( ) sin( ) *T bT P t T P Ce e A T A T <sup>s</sup>*

<sup>3</sup> 3 3 ( ) 1 11 1 32 3 ( ) cos( ) sin( ) *T T T bT Pe P A e e A T A T*

We now solve equations (4, 5 & 6), to determine the unknown parameters *ma, Rp* (and *T2*).

λ λ

<sup>2</sup> ( )

ω

λ

• At *t =T3, Ps(t =T3) = Pd(t =T3).* Hence, from equations (2 and 3), we obtain:

 λ

2 1 3 24 2 ( )( ) cos( ) sin( ) 0 *dP t <sup>s</sup> T bT tT Ce e A T A T*

ω

( 1)

λ

ω

λ

2 2

ω

[( ) ]

 ω

λ

ω

− − = == + <sup>⎡</sup> <sup>+</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> (5)

ω

− −− =− + <sup>⎡</sup> <sup>+</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> (6)

ωω− − = =− + ⎡ ⎤ <sup>−</sup> <sup>=</sup> ⎣ ⎦ (4)

> ω

> > ω

(3)

phase is from *T3* to *T4* [13]. • During systolic phase,

Also, as noted in Fig 28, the boundary values are:

• At *t =T2, Ps =P2*. Hence, from equations (3 & 4), we get:

*dt*

λ

where

We have determined the expressions for the aortic pressure during the systolic and diastolic phases, by solving the governing equation (1), for the monitored LV outflow rate (or input into the aorta) *I(t)*, using (i) the monitored auscultatory diastolic pressures (*Pad*), to serve as the boundary condition at the beginning of the systolic-phase solution (at time *T1*) and at the end of the diastolic-phase solution (at time *T4*), (ii) the monitored auscultatory systolic pressure (*Pas =P2*) to represent the maximum value of the systolic-phase solution.

Because the pressure solution of equation (1) is a function of *ma* and *Rp*, we first determine the values of these parameters by making the solution satisfy the above-mentioned boundary conditions, which in turn yields the pressure profile given by *Pd(t)* and *Ps(t).* This noninvasively determinable aortic pressure can provide hitherto unavailable information on vascular compliance and resistance status, as well as on the capacity of the LV to respond to it.

Fig. 29. Plot of aortic pressure (of patient B) during one cardiac cycle, *te*=0.32s. Herein, *T1-T3* represents the systolic phase, and *T3-T4(=0.92s)* represents the diastolic phase. At *T2*, the aortic systolic pressure profile has its maximal value *(=Ps*). The scatter points are the data measured from catheterization. The solid line is the model-computed profile; RMS=2.41 mmHg. Note the excellent correlation between the model-derived aortic pressure profile and the catheter-obtained aortic pressure profile.

In **Fig. 29**, the model-computed aortic pressure profile patient B (with double vessel disease and hypertension) is shown, along with the actual catheter pressure data. We can note how well the model-computed result matches the actual catheterization data, with RMS 2.41 mmHg. The aortic stiffness (*ma*) and peripheral resistance (*Rp*) are obtained to be 1.03 *mmHg/ml* and 1.59 *mmHg·s/ml*, respectively.

Let us consider yet another benefit of this analysis. We have determined aortic pressure profile, aortic stiffness (*ma*) or aortic elastance (*Eao*) and peripheral resistance. From the instantaneous aortic pressure and aortic inflow rate, we can determine the instantaneous left ventricular (LV) systolic pressure, in terms of the instantaneous dimensions of the LV outflow tract. We hence determine the LV systolic pressure profile, from which we can evaluate the traditional contractility index (*dP/dt max*) as well as the LV systolic elastance (*Elv*).

We can then determine the ratio of *Elv/Eao*, to represent the LV-Aorta Matching Index (*VAMI*). In ischemic cardiomyopathy patients, this *VAMI* value is depressed. However, following surgical vascular restoration, this index value is partially restored.

### **11. Coronary Bypass surgery: Candidacy**

• **Interventional candidacy based on computed intra-LV flow-velocity and pressuregradient distributions** 

Fig. 30. Construction of Intra-LV Blood flow velocity and pressure-gradient distributions for a patient with myocardial infarct: (a) Superimposed sequential diastolic and systolic endocardial frames (whose aortic valves centers and the long axis are matched), before (1) and after (2) administration of nitroglycerin; (b) Instantaneous intra-LV distributions of velocity during diastole, before (1) and after (2) administration of nitroglycerin; (c) Instantaneous intra-LV distributions of velocity during ejection phase, before (1) and after (2) administration of nitroglycerin, (d) Instantaneous intra-LV distributions of pressure-differentials during diastole, before (1) and after (2) administration of nitroglycerin; (e) Instantaneous intra-LV distributions of pressure-differential during ejection phase, before (1) and after (2) administration of nitroglycerin. (Adapted from reference 14: Subbaraj K, Ghista DN, Fallen EL, *J Biomed Eng* 1987, 9:206-215.).

• **Interventional candidacy based on computed intra-LV flow-velocity and pressure-**

Fig. 30. Construction of Intra-LV Blood flow velocity and pressure-gradient distributions for

a patient with myocardial infarct: (a) Superimposed sequential diastolic and systolic endocardial frames (whose aortic valves centers and the long axis are matched), before (1) and after (2) administration of nitroglycerin; (b) Instantaneous intra-LV distributions of

(c) Instantaneous intra-LV distributions of velocity during ejection phase, before (1) and after (2) administration of nitroglycerin, (d) Instantaneous intra-LV distributions of pressure-differentials during diastole, before (1) and after (2) administration of nitroglycerin; (e) Instantaneous intra-LV distributions of pressure-differential during ejection phase, before (1) and after (2) administration of nitroglycerin. (Adapted from reference 14: Subbaraj K, Ghista DN, Fallen EL, *J Biomed Eng* 1987, 9:206-215.).

velocity during diastole, before (1) and after (2) administration of nitroglycerin;

**11. Coronary Bypass surgery: Candidacy** 

**gradient distributions** 

A left ventricle with ischemic and infarcted myocardial segments will have lowered ejection fraction and cardiac output, because it will not be able to generate adequate myocardial contraction to raise the intra-LV pressure above aortic pressure for a ling enough duration to generate adequate stroke volume. These patients need coronary bypass surgery, and a presurgical assessment of their candidacy for it, on how much they can from it. For this purpose, we need to determine the intra-LV blood flow velocity and pressure-gradient profiles before and after the administration of nitroglycerin.

So, we carry out a CFD analysis of intra-LV blood flow. The data required for the CFD analysis consists of: LV 2-D long-axis frames during LV diastolic and systolic phases; LV pressure vs. time associated with these LV frames; Computation of LV instantaneous wall velocities as well as instantaneous velocity of blood entering the LV during the filling phase and leaving the LV during the ejection phase.

From this CFD, we have determined the instantaneous distributions of intra-LV blood-flow velocity and differential-pressure during filling and ejection phases, to intrinsically characterize LV resistance-to-filling (LV-RFT) and contractility (LV-CONT) respectively. The results are summarized in the above Fig. 30**.**

By comparing intra-LV pressure-gradients before and after administration of nitroglycerin (a myocardial perfusing agent, and hence a quasi-simulator of coronary bypass surgery), we can infer how the myocardium is going to respond and how these LV functional indices will improve after coronary by bypass surgery.

## **12. Theory of hospital administration: Formulation of hospital units' performance index and cost-effective index**

A Hospital has clinical services departments, medical supply and hospital-services departments, financial-management and administrative departments. Each of these five sets of departments has to function in a cost-effective fashion.

**Let us, for example, consider the Intensive-Care Unit (ICU) department.** The human resource to an ICU dept consists of physicians and nurses. Using activity-based costing, we can determine the human-resource strength, based on an assumed reasonable probabilityof-occurrence of (for instance) two patients simultaneously (instead of just one patient) having life-threatening episodes.

**Performance Index:** We can formulate the ICU performance-indicator in terms of the amounts by which the physiological health-index (PHI or NDPI) values of patients were (i) enhanced in the ICU for those patients discharged into the ward from the ICU, and (ii) diminished in the ICU in the case of patients who died in the ICU.

Let us say that patients are admitted to the CCU if their Physiological-health-index (PHI) value falls below 50%. So if the PHI of a patient improves from 30 to 50, then the **Patient-HealthImprovement Index (PHII)** for that patient is given by [15],

$$\text{PHIII} = \left(\frac{50 - 30}{30}\right) \cdot 100 = 67 \text{ (or } 67\%\text{)}\tag{1}$$

Thus the **patient health-improvement index (PHII)** value is higher if a more seriously-ill patient is discharged from the ICU, and lower if a not-so-seriously ill patient is discharged, i.e., if

$$\text{PPIII} = \left(\frac{50 - 40}{40}\right) \text{100} = 25 \text{ (or } 25\%\text{)}\tag{2}$$

We can then formulate the **Performance-index (PFI)** for an ICU as follows:

$$MCI \text{ Performance index (PFI)} = \frac{\sum \text{PHI of the patients}}{\text{\# of those patients treated during a life time period}} \tag{3}$$

Hence, the higher the value of ICU performance index, the better is the performance of the ICU. If now a patient dies, as a result of the PHII becoming negative (i.e slipping from (say) 30 to 10), then

$$PHII = \ 100 \ \left(\frac{10-30}{30}\right) = \ -67\tag{4}$$

As a result, ΣPHII (in equation 3) will decrease, and the overall value of ICU performance index (namely PFI, as calculated by means of eq. 3), will fall.

**Cost-Effective Index:** Now, let us consider that (i) we have one physician and five nurses for a 10 bed CCU, based on the probability-of-occurrence of two patients having lifethreatening events being say 0.2 (or 20%), and that (ii) for this human resource/staffing, the ICU performance index value (PFI) is (say) 40.

If we increase the staffing, the ICU performance index value could go upto 50 or so, at the expense of more salary cost. So now we can come up with another indicator namely, **Costeffectiveness index (CEI)**

$$\text{CEI} = \frac{\text{Performance index}}{\text{Total salary index (in salary - units)}}$$

$$= \frac{\text{Performance index}}{\text{Ressource index (in terms of salary - units)}} \tag{1}$$

(5)

wherein, say, a salary of 1000=0.1 unit, 10,000=1 unit, 20,000=2 units, and so on. So if an ICU has one physician with a monthly salary of 20,000 (i.e. 2 salary-units) and four nurses each with a total monthly salary of 5,000 (i.e. total of 2.0 salary-units), then

$$\begin{aligned} \text{CEI(ICU)} &= \frac{\text{Performance index (of40)}}{\text{Solary-units Index or Resistance Index}, \text{R}\_i [= (2 + 2.0)]} \\ &= \frac{40}{4.0} = 10 \end{aligned} \tag{6}$$

Now, let us assume that we raise the **PFI** (ICU) to (say) 60 by augmenting the nursing staff, so as to have six nurses (Ri = 3 units) and 1.5 full-time equivalent physician-on-duty (Ri = 3 units), then

$$\text{CEI (ICLI)} = \frac{PFI}{R\_i} = \frac{50}{(3+3)} = 8.3\tag{7}$$

So while the PFI of ICU has gone up from 40 to 50, the CEI of ICU has gone down from 10 to 8.3.

**Strategy of Operation:** Our strategy would be to operate this Performance-Resource system, in such a way that we can determine the resource index Ri for which we can obtain acceptable values of both PFI and CEI. In a way, figure (31) could represent this balance between PFI & CEI, in order to determine optimal Resource Index or resource [15].

With reference to this **figure 31**, if we have a resource value of Ri = R1, then the corresponding PFI (=PFI1) will be unacceptable, as being too low; hence, we will want to increase the value of Ri. If we have a resource value of Ri = R2, then our corresponding CEI (= CEI2) will also be unacceptable, for being too low; hence we will want to decrease Ri. In doing so, we can arrive at the optimal value Rio, for which both CEI and PFI are acceptable. This procedure, for converging to *Ri = Ri0,* can be formulated computationally.

Fig. 31. Optimising the value of Ri so to obtain acceptable values of CEI and PFI [15].

**Now then, let us formulate how a hospital budget can be optimally distributed**. Let us say that a Hospital has 'n' number of departments and a prescribed budget (or budget index, BGI). We would want to distribute the budget among the departments, such that none of the 'n' departments has a PFI below the acceptable value of PFIa and a CEI below the acceptable value of CEIa**.** 
