The series resistors can be used as current sensors. For each LED branch, a PNP transistor's E-B junction is monitoring the voltage across the series resistor. The PNPs are placed in series and are used to maintain a positive gate voltage on the NMOS M1 used as a primary switch.
The circuits below use easily available parts and are inexpensive.
Burn-Out Detector
SW1 is used to initially turn the LEDs ON.
If any LED goes open, the current sense voltage goes to 0, the associated PNP transistor opens, and the gate is pulled to 0V via R7.
If an "ON" pushbutton is not desired, it can be replaced with a 100nF capacitor. That way the circuit will start up with the LEDs turned ON.
![schematic](https://cdn.statically.io/img/i.sstatic.net/5L1UX.png)
simulate this circuit – Schematic created using CircuitLab
The plot below shows the total LED current. SW1 is depressed from 0ms to 1ms. At 5ms, SW3 goes open, simulating a failure of the 3rd branch. The LEDs then turn off.
![The LED current waveform](https://cdn.statically.io/img/i.sstatic.net/3T7G2.png)
Voltage Regulator with Burn-Out Detector
It takes only a small modification to the circuit to use the pass element M1 to both turn the LEDs off when one burns out, but also to regulate the voltage.
![schematic](https://cdn.statically.io/img/i.sstatic.net/6a7M2.png)
simulate this circuit
The voltage regulation is not too shabby for a three-transistor circuit. R10 should is used to adjust the voltage.
![The output-vs-input voltage regulation plot](https://cdn.statically.io/img/i.sstatic.net/fynsT.png)
The gate- and LED-supply voltage throughout the lifecycle of the system are shown below, for supply voltages of 11.5, 12, 12.5 and 30V. From 0ms to 1ms, the ON button is depressed. At 5ms, SW4 opens, simulating a LED failure in the 3rd string.
![The LED voltage waveforms for supply voltages of 11.5, 12, 12.5 and 30V](https://cdn.statically.io/img/i.sstatic.net/F2Qzt.png)
One-Branch Current Regulator with Burn-Out Detector
This circuit, instead of regulating voltage, regulates current in one of the LED branches. The remaining branches are assumed to be at a similar temperature and will draw similar current. The current regulation is very good.
![schematic](https://cdn.statically.io/img/i.sstatic.net/oUBWl.png)
simulate this circuit
The LED current for supply voltages of 11V, 12.5V and 30V is plotted below. The ON switch is depressed from 0ms to 1ms, and the 3rd LED branch opens at 5ms.
![The LED current waveforms for the 11, 12.5 and 30V supply voltages](https://cdn.statically.io/img/i.sstatic.net/17KhN.png)
All-Branch Current Regulator with Burn-Out Detection
Instead of using just one branch as an "indicator" of current, the average current across all branches can be regulated. This will compensate better for unequal LED aging, etc.
![schematic](https://cdn.statically.io/img/i.sstatic.net/D2cta.png)
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Again, the current regulation is decent:
![The LED current waveforms for the 11, 12.5 and 30V supply voltages](https://cdn.statically.io/img/i.sstatic.net/Tjnqo.png)